Let's continue the conversation about actions with matrices. Namely, during the study of this lecture you will learn how to find the inverse matrix. Learn. Even if math is difficult.
What is an inverse matrix? Here we can draw an analogy with reciprocal numbers: Consider, for example, the optimistic number 5 and its inverse number . The product of these numbers is equal to one: . Everything is similar with matrices! The product of a matrix and its inverse matrix is equal to – identity matrix, which is the matrix analogue of the numerical unit. However, first things first – let’s solve the important one first. practical question, namely, we will learn how to find this very inverse matrix.
What do you need to know and be able to do to find the inverse matrix? You must be able to decide qualifiers. You must understand what it is matrix and be able to perform some actions with them.
There are two main methods for finding the inverse matrix:
by using algebraic additions And using elementary transformations.
Today we will study the first, simpler method.
Let's start with the most terrible and incomprehensible. Let's consider square matrix. The inverse matrix can be found using the following formula:
Where is the determinant of the matrix, is the transposed matrix of algebraic complements of the corresponding elements of the matrix.
The concept of an inverse matrix exists only for square matrices, matrices “two by two”, “three by three”, etc.
Designations: As you may have already noticed, the inverse matrix is denoted by a superscript
Let's start with the simplest case - a two-by-two matrix. Most often, of course, “three by three” is required, but, nevertheless, I strongly recommend studying a simpler task in order to master general principle solutions.
Example:
Find the inverse of a matrix
Let's decide. It is convenient to break down the sequence of actions point by point.
1) First we find the determinant of the matrix.
If your understanding of this action is not good, read the material How to calculate the determinant?
Important! If the determinant of the matrix is equal to ZERO– inverse matrix DOES NOT EXIST.
In the example under consideration, as it turned out, , which means everything is in order.
2) Find the matrix of minors.
To solve our problem, it is not necessary to know what a minor is, however, it is advisable to read the article How to calculate the determinant.
The matrix of minors has the same dimensions as the matrix, that is, in this case.
The only thing left to do is find four numbers and put them instead of asterisks.
Let's return to our matrix
Let's look at the top left element first:
How to find it minor?
And this is done like this: MENTALLY cross out the row and column in which this element is located:
The remaining number is minor of this element, which we write in our matrix of minors:
Consider the following matrix element:
Mentally cross out the row and column in which this element appears:
What remains is the minor of this element, which we write in our matrix:
Similarly, we consider the elements of the second row and find their minors:
Ready.
It's simple. In the matrix of minors you need CHANGE SIGNS two numbers:
These are the numbers that I circled!
– matrix of algebraic additions of the corresponding elements of the matrix.
And just...
4) Find the transposed matrix of algebraic additions.
– transposed matrix of algebraic complements of the corresponding elements of the matrix.
5) Answer.
Let's remember our formula
Everything has been found!
So the inverse matrix is: 
It is better to leave the answer as is. NO NEED divide each element of the matrix by 2, as you get fractional numbers. This nuance is discussed in more detail in the same article. Actions with matrices.
How to check the solution?
You need to perform matrix multiplication or
Examination: 
Received already mentioned identity matrix is a matrix with ones by main diagonal and zeros in other places.
Thus, the inverse matrix is found correctly.
If you carry out the action, the result will also be an identity matrix. This is one of the few cases where matrix multiplication is permutable, more detailed information can be found in the article Properties of operations on matrices. Matrix Expressions. Also note that during the check, the constant (fraction) is brought forward and processed at the very end - after the matrix multiplication. This is a standard technique.
Let's move on to a more common case in practice - the three-by-three matrix:
Example:
Find the inverse of a matrix 
The algorithm is exactly the same as for the “two by two” case.
We find the inverse matrix using the formula: , where is the transposed matrix of algebraic complements of the corresponding elements of the matrix.
1) Find the determinant of the matrix.
Here the determinant is revealed on the first line.
Also, don’t forget that, which means everything is fine - inverse matrix exists.
2) Find the matrix of minors.
The matrix of minors has a dimension of “three by three” 
, and we need to find nine numbers.
I'll look at a couple of minors in detail:
Consider the following matrix element: 
MENTALLY cross out the row and column in which this element is located: 
We write the remaining four numbers in the “two by two” determinant. 
This two-by-two determinant and is the minor of this element. It needs to be calculated: 
That’s it, the minor has been found, we write it in our matrix of minors: 
As you probably guessed, you need to calculate nine two-by-two determinants. The process, of course, is tedious, but the case is not the most severe, it can be worse.
Well, to consolidate – finding another minor in the pictures: 
Try to calculate the remaining minors yourself.
Final result: 
– matrix of minors of the corresponding elements of the matrix.
The fact that all the minors turned out to be negative is purely an accident.
3) Find the matrix of algebraic additions.
In the matrix of minors it is necessary CHANGE SIGNS strictly for the following elements: 
In this case:
We do not consider finding the inverse matrix for a “four by four” matrix, since such a task can only be given by a sadistic teacher (for the student to calculate one “four by four” determinant and 16 “three by three” determinants). In my practice, there was only one such case, and the customer of the test paid quite dearly for my torment =).
In a number of textbooks and manuals you can find a slightly different approach to finding the inverse matrix, but I recommend using the solution algorithm outlined above. Why? Because the likelihood of getting confused in calculations and signs is much less.
The matrix $A^(-1)$ is called the inverse of the square matrix $A$ if the condition $A^(-1)\cdot A=A\cdot A^(-1)=E$ is satisfied, where $E $ is the identity matrix, the order of which is equal to the order of the matrix $A$.
A non-singular matrix is a matrix whose determinant is not equal to zero. Accordingly, a singular matrix is one whose determinant is equal to zero.
The inverse matrix $A^(-1)$ exists if and only if the matrix $A$ is non-singular. If the inverse matrix $A^(-1)$ exists, then it is unique.
There are several ways to find the inverse of a matrix, and we will look at two of them. This page will discuss the adjoint matrix method, which is considered standard in most higher mathematics courses. The second method of finding the inverse matrix (the method of elementary transformations), which involves using the Gauss method or the Gauss-Jordan method, is discussed in the second part.
Adjoint matrix method
Let the matrix $A_(n\times n)$ be given. In order to find the inverse matrix $A^(-1)$, three steps are required:
- Find the determinant of the matrix $A$ and make sure that $\Delta A\neq 0$, i.e. that matrix A is non-singular.
- Compose algebraic complements $A_(ij)$ of each element of the matrix $A$ and write the matrix $A_(n\times n)^(*)=\left(A_(ij) \right)$ from the found algebraic complements.
- Write the inverse matrix taking into account the formula $A^(-1)=\frac(1)(\Delta A)\cdot (A^(*))^T$.
The matrix $(A^(*))^T$ is often called adjoint (reciprocal, allied) to the matrix $A$.
If the solution is done manually, then the first method is good only for matrices of relatively small orders: second (), third (), fourth (). To find the inverse of a higher order matrix, other methods are used. For example, the Gaussian method, which is discussed in the second part.
Example No. 1
Find the inverse of matrix $A=\left(\begin(array) (cccc) 5 & -4 &1 & 0 \\ 12 &-11 &4 & 0 \\ -5 & 58 &4 & 0 \\ 3 & - 1 & -9 & 0 \end(array) \right)$.
Since all elements of the fourth column are equal to zero, then $\Delta A=0$ (i.e. the matrix $A$ is singular). Since $\Delta A=0$, there is no inverse matrix to matrix $A$.
Example No. 2
Find the inverse of matrix $A=\left(\begin(array) (cc) -5 & 7 \\ 9 & 8 \end(array)\right)$.
We use the adjoint matrix method. First, let's find the determinant of the given matrix $A$:
$$ \Delta A=\left| \begin(array) (cc) -5 & 7\\ 9 & 8 \end(array)\right|=-5\cdot 8-7\cdot 9=-103. $$
Since $\Delta A \neq 0$, then the inverse matrix exists, therefore we will continue the solution. Finding algebraic complements
\begin(aligned) & A_(11)=(-1)^2\cdot 8=8; \; A_(12)=(-1)^3\cdot 9=-9;\\ & A_(21)=(-1)^3\cdot 7=-7; \; A_(22)=(-1)^4\cdot (-5)=-5.\\ \end(aligned)
We compose a matrix of algebraic additions: $A^(*)=\left(\begin(array) (cc) 8 & -9\\ -7 & -5 \end(array)\right)$.
We transpose the resulting matrix: $(A^(*))^T=\left(\begin(array) (cc) 8 & -7\\ -9 & -5 \end(array)\right)$ (the resulting matrix is often is called the adjoint or allied matrix to the matrix $A$). Using the formula $A^(-1)=\frac(1)(\Delta A)\cdot (A^(*))^T$, we have:
$$ A^(-1)=\frac(1)(-103)\cdot \left(\begin(array) (cc) 8 & -7\\ -9 & -5 \end(array)\right) =\left(\begin(array) (cc) -8/103 & 7/103\\ 9/103 & 5/103 \end(array)\right) $$
So, the inverse matrix is found: $A^(-1)=\left(\begin(array) (cc) -8/103 & 7/103\\ 9/103 & 5/103 \end(array)\right) $. To check the truth of the result, it is enough to check the truth of one of the equalities: $A^(-1)\cdot A=E$ or $A\cdot A^(-1)=E$. Let's check the equality $A^(-1)\cdot A=E$. In order to work less with fractions, we will substitute the matrix $A^(-1)$ not in the form $\left(\begin(array) (cc) -8/103 & 7/103\\ 9/103 & 5/103 \ end(array)\right)$, and in the form $-\frac(1)(103)\cdot \left(\begin(array) (cc) 8 & -7\\ -9 & -5 \end(array )\right)$:
Answer: $A^(-1)=\left(\begin(array) (cc) -8/103 & 7/103\\ 9/103 & 5/103 \end(array)\right)$.
Example No. 3
Find the inverse matrix for the matrix $A=\left(\begin(array) (ccc) 1 & 7 & 3 \\ -4 & 9 & 4 \\ 0 & 3 & 2\end(array) \right)$.
Let's start by calculating the determinant of the matrix $A$. So, the determinant of the matrix $A$ is:
$$ \Delta A=\left| \begin(array) (ccc) 1 & 7 & 3 \\ -4 & 9 & 4 \\ 0 & 3 & 2\end(array) \right| = 18-36+56-12=26. $$
Since $\Delta A\neq 0$, then the inverse matrix exists, therefore we will continue the solution. We find the algebraic complements of each element of a given matrix:
We compose a matrix of algebraic additions and transpose it:
$$ A^*=\left(\begin(array) (ccc) 6 & 8 & -12 \\ -5 & 2 & -3 \\ 1 & -16 & 37\end(array) \right); \; (A^*)^T=\left(\begin(array) (ccc) 6 & -5 & 1 \\ 8 & 2 & -16 \\ -12 & -3 & 37\end(array) \right) $$
Using the formula $A^(-1)=\frac(1)(\Delta A)\cdot (A^(*))^T$, we get:
$$ A^(-1)=\frac(1)(26)\cdot \left(\begin(array) (ccc) 6 & -5 & 1 \\ 8 & 2 & -16 \\ -12 & - 3 & 37\end(array) \right)= \left(\begin(array) (ccc) 3/13 & -5/26 & 1/26 \\ 4/13 & 1/13 & -8/13 \ \ -6/13 & -3/26 & 37/26 \end(array) \right) $$
So $A^(-1)=\left(\begin(array) (ccc) 3/13 & -5/26 & 1/26 \\ 4/13 & 1/13 & -8/13 \\ - 6/13 & -3/26 & 37/26 \end(array) \right)$. To check the truth of the result, it is enough to check the truth of one of the equalities: $A^(-1)\cdot A=E$ or $A\cdot A^(-1)=E$. Let's check the equality $A\cdot A^(-1)=E$. In order to work less with fractions, we will substitute the matrix $A^(-1)$ not in the form $\left(\begin(array) (ccc) 3/13 & -5/26 & 1/26 \\ 4/13 & 1/13 & -8/13 \\ -6/13 & -3/26 & 37/26 \end(array) \right)$, and in the form $\frac(1)(26)\cdot \left( \begin(array) (ccc) 6 & -5 & 1 \\ 8 & 2 & -16 \\ -12 & -3 & 37\end(array) \right)$:
The check was successful, the inverse matrix $A^(-1)$ was found correctly.
Answer: $A^(-1)=\left(\begin(array) (ccc) 3/13 & -5/26 & 1/26 \\ 4/13 & 1/13 & -8/13 \\ -6 /13 & -3/26 & 37/26 \end(array) \right)$.
Example No. 4
Find the matrix inverse of matrix $A=\left(\begin(array) (cccc) 6 & -5 & 8 & 4\\ 9 & 7 & 5 & 2 \\ 7 & 5 & 3 & 7\\ -4 & 8 & -8 & -3 \end(array) \right)$.
For a fourth-order matrix, finding the inverse matrix using algebraic additions is somewhat difficult. However, such examples do occur in test papers.
To find the inverse of a matrix, you first need to calculate the determinant of the matrix $A$. The best way to do this in this situation is by decomposing the determinant along a row (column). Select any row or column and find algebraic additions each element of the selected row or column.
Matrix algebra - Inverse matrixinverse matrix
Inverse matrix is called a matrix which, when multiplied both on the right and on the left by this matrix gives the identity matrix.
Let us denote the inverse matrix of the matrix A through , then according to definition we get:
Where E– identity matrix.
Square matrix called not special (non-degenerate) if its determinant is not zero. Otherwise it is called special (degenerate) or singular.
The theorem holds: Every non-singular matrix has an inverse matrix.
The operation of finding the inverse matrix is called appeal matrices. Let's consider the matrix inversion algorithm. Let a non-singular matrix be given n-th order:
where Δ = det A ≠ 0.
Algebraic addition of an element matrices n-th order A is called the determinant of a matrix taken with a certain sign ( n–1)th order obtained by deleting i-th line and j th matrix column A:
Let's create the so-called attached matrix:
where are the algebraic complements of the corresponding elements of the matrix A.
Note that algebraic additions of matrix row elements A are placed in the corresponding columns of the matrix Ã
, that is, the matrix is transposed at the same time.
By dividing all the elements of the matrix Ã
by Δ – the value of the matrix determinant A, we get the inverse matrix as a result:
Let's note the row special properties inverse matrix:
1) for a given matrix A its inverse matrix
is the only one;
2) if there is an inverse matrix, then right reverse And left reverse the matrices coincide with it;
3) a singular (singular) square matrix does not have an inverse matrix.
Basic properties of an inverse matrix:
1) the determinant of the inverse matrix and the determinant of the original matrix are reciprocals;
2) the inverse matrix of the product of square matrices is equal to the product of the inverse matrix of factors, taken in reverse order:
3) the transposed inverse matrix is equal to the inverse matrix of the given transposed matrix:
EXAMPLE Calculate the inverse of the given matrix.
Matrix A -1 is called the inverse matrix with respect to matrix A if A*A -1 = E, where E is the identity matrix of the nth order. An inverse matrix can only exist for square matrices.
Purpose of the service. By using of this service online you can find algebraic complements, transposed matrix A T, allied matrix and inverse matrix. The decision is carried out directly on the website (online) and is free. The calculation results are presented in a report in Word and Excel format (i.e., it is possible to check the solution). see design example.
Instructions. To obtain a solution, it is necessary to specify the dimension of the matrix. Next, fill out matrix A in the new dialog box.
See also Inverse matrix using the Jordano-Gauss method
Algorithm for finding the inverse matrix
- Finding the transposed matrix A T .
- Definition of algebraic complements. Replace each element of the matrix with its algebraic complement.
- Compiling an inverse matrix from algebraic additions: each element of the resulting matrix is divided by the determinant of the original matrix. The resulting matrix is the inverse of the original matrix.
- Determine whether the matrix is square. If not, then there is no inverse matrix for it.
- Calculation of the determinant of the matrix A. If it is not equal to zero, we continue the solution, otherwise the inverse matrix does not exist.
- Definition of algebraic complements.
- Filling out the union (mutual, adjoint) matrix C .
- Compiling an inverse matrix from algebraic additions: each element of the adjoint matrix C is divided by the determinant of the original matrix. The resulting matrix is the inverse of the original matrix.
- They do a check: they multiply the original and the resulting matrices. The result should be an identity matrix.
Example No. 1. Let's write the matrix in the form:
Algebraic additions.
| A 1,1 = (-1) 1+1 |
|
∆ 1,1 = (-1 4-5 (-2)) = 6
| A 1,2 = (-1) 1+2 |
|
∆ 1,2 = -(2 4-(-2 (-2))) = -4
| A 1.3 = (-1) 1+3 |
|
∆ 1,3 = (2 5-(-2 (-1))) = 8
| A 2,1 = (-1) 2+1 |
|
∆ 2,1 = -(2 4-5 3) = 7
| A 2,2 = (-1) 2+2 |
|
∆ 2,2 = (-1 4-(-2 3)) = 2
| A 2,3 = (-1) 2+3 |
|
∆ 2,3 = -(-1 5-(-2 2)) = 1
| A 3.1 = (-1) 3+1 |
|
∆ 3,1 = (2 (-2)-(-1 3)) = -1
| A 3.2 = (-1) 3+2 |
|
∆ 3,2 = -(-1 (-2)-2 3) = 4
| A 3.3 = (-1) 3+3 |
|
∆ 3,3 = (-1 (-1)-2 2) = -3
Then inverse matrix can be written as:
| A -1 = 1/10 |
|
| A -1 = |
|
Another algorithm for finding the inverse matrix
Let us present another scheme for finding the inverse matrix.- Find the determinant of this square matrix A.
- We find algebraic complements to all elements of the matrix A.
- We write algebraic additions of row elements to columns (transposition).
- We divide each element of the resulting matrix by the determinant of the matrix A.
A special case: The inverse of the identity matrix E is the identity matrix E.
Typically, inverse operations are used to simplify complex algebraic expressions. For example, if the problem involves the operation of dividing by a fraction, you can replace it with the operation of multiplying by the reciprocal of a fraction, which is the inverse operation. Moreover, matrices cannot be divided, so you need to multiply by the inverse matrix. Calculating the inverse of a 3x3 matrix is quite tedious, but you need to be able to do it manually. You can also find the reciprocal using a good graphing calculator.
Steps
Using the adjoint matrix
Transpose the original matrix. Transposition is the replacement of rows with columns relative to the main diagonal of the matrix, that is, you need to swap the elements (i,j) and (j,i). In this case, the elements of the main diagonal (starts in the upper left corner and ends in the lower right corner) do not change.
- To change rows to columns, write the elements of the first row in the first column, the elements of the second row in the second column, and the elements of the third row in the third column. The order of changing the position of the elements is shown in the figure, in which the corresponding elements are circled with colored circles.
Find the definition of each 2x2 matrix. Every element of any matrix, including a transposed one, is associated with a corresponding 2x2 matrix. To find a 2x2 matrix that corresponds to a specific element, cross out the row and column in which the given element is located, that is, you need to cross out five elements of the original 3x3 matrix. Four elements will remain uncrossed, which are elements of the corresponding 2x2 matrix.
- For example, to find a 2x2 matrix for the element that is located at the intersection of the second row and the first column, cross out the five elements that are in the second row and first column. The remaining four elements are elements of the corresponding 2x2 matrix.
- Find the determinant of each 2x2 matrix. To do this, subtract the product of the elements of the secondary diagonal from the product of the elements of the main diagonal (see figure).
- Detailed information about 2x2 matrices corresponding to specific elements of a 3x3 matrix can be found on the Internet.
Create a cofactor matrix. Write the results obtained earlier in the form of a new cofactor matrix. To do this, write the found determinant of each 2x2 matrix where the corresponding element of the 3x3 matrix was located. For example, if you are considering a 2x2 matrix for element (1,1), write its determinant in position (1,1). Then change the signs of the corresponding elements according to a certain scheme, which is shown in the figure.
- Scheme for changing signs: the sign of the first element of the first line does not change; the sign of the second element of the first line is reversed; the sign of the third element of the first line does not change, and so on line by line. Please note that the “+” and “-” signs that are shown in the diagram (see figure) do not indicate that the corresponding element will be positive or negative. In this case, the “+” sign indicates that the sign of the element does not change, and the “-” sign indicates a change in the sign of the element.
- Detailed information about cofactor matrices can be found on the Internet.
- This way you will find the adjoint matrix of the original matrix. It is sometimes called a complex conjugate matrix. Such a matrix is denoted as adj(M).
Divide each element of the adjoint matrix by its determinant. The determinant of the matrix M was calculated at the very beginning to check that the inverse matrix exists. Now divide each element of the adjoint matrix by this determinant. Write the result of each division operation where the corresponding element is located. This way you will find the matrix inverse to the original one.
- The determinant of the matrix which is shown in the figure is 1. Thus, here the adjoint matrix is the inverse matrix (because when any number is divided by 1, it does not change).
- In some sources, the division operation is replaced by the operation of multiplication by 1/det(M). However, the final result does not change.
Write the inverse matrix. Write the elements located on the right half of the large matrix as a separate matrix, which is the inverse matrix.
Enter the original matrix into the calculator's memory. To do this, click the Matrix button, if available. For a Texas Instruments calculator, you may need to press the 2nd and Matrix buttons.
Select the Edit menu. Do this using the arrow buttons or the appropriate function button located at the top of the calculator's keyboard (the location of the button varies depending on the calculator model).
Enter the matrix notation. Most graphic calculators can work with 3-10 matrices, which can be designated letters A-J. Typically, just select [A] to designate the original matrix. Then press the Enter button.
Enter the matrix size. This article talks about 3x3 matrices. But graphic calculators can work with matrices large sizes. Enter the number of rows, press Enter, then enter the number of columns and press Enter again.
Enter each matrix element. A matrix will be displayed on the calculator screen. If you have previously entered a matrix into the calculator, it will appear on the screen. The cursor will highlight the first element of the matrix. Enter the value for the first element and press Enter. The cursor will automatically move to next element matrices.