The inverse matrix or invertible matrix is ​​a type of square matrix, that is, it has the same number of rows (m) and columns (n).

It occurs when the product of two matrices results in a identity matrix of the same order (same number of rows and columns).

Therefore, to find the inverse of a matrix, multiplication is used.

A. B = B. A = In (when matrix B is inverse of matrix A)

But what is the identity matrix?

The identity matrix is ​​defined when the main diagonal elements are all equal to 1 and the other elements are equal to 0 (zero). It is indicated by In:

Inverse matrix properties

  • There is only one inverse for each matrix
  • Not all matrices have an inverse matrix. It is invertible only when the products of the square matrices result in an identity matrix (In)
  • The inverse matrix of an inverse corresponds to the matrix itself: A = (A-1)-1
  • The transpose matrix of an inverse matrix is ​​also inverse: (At) -1 = (A-1)t
  • The inverse matrix of a transpose matrix corresponds to the inverse transposition: (A-1 Unt)-1
  • The inverse matrix of an identity matrix is ​​the same as the identity matrix: I-1 = I

see also: Arrays

Inverse matrix examples

2 × 2 inverse matrix

3 × 3 inverse matrix

Step by step: how to calculate the inverse matrix?

We know that if the product of two matrices is equal to the identity matrix, that matrix has an inverse.

Note that if matrix A is inverse of matrix B, the notation: A-1.

Example: Find the inverse of the matrix under the order 3 × 3.

First of all, we must remember that. A-1 = I (The matrix multiplied by its inverse will result in the identity matrix In)

Each element in the first row of the first matrix is ​​multiplied by each column of the second matrix.

Therefore, the elements of the second row of the first matrix are multiplied by the columns of the second.

And finally, the third row of the first with the columns of the second:

By equivalence of the elements with the identity matrix, we can discover the values ​​of:

a = 1
b = 0
c = 0

inverse matrix properties

Knowing these values, we can calculate the other unknowns in the matrix. In the third row and the first column of the first matrix we have a + 2d = 0. So, let's start by finding the value of d, by replacing the found values:

1 + 2d = 0
2d = -1
d = -1/2

Similarly, in the third row and the second column we can find the value of y:

b + 2e = 0
0 + 2e = 0
2e = 0
e = 0/2
e = 0

Moving on, we have in the third row of the third column: c + 2f. Note that, secondly, the identity matrix of this equation is not equal to zero, but equal to 1.

c + 2f = 1
0 + 2f = 1
2f = 1
f = ½

Going to the second row and the first column, we will find the value of g:

a + 3d + g = 0
1 + 3. (-1/2) + g = 0
1 - 3/2 + g = 0
g = -1 + 3/2
g = ½

In the second row and the second column, we can find the value of h:

b + 3e + h = 1
0 + 3. 0 + h = 1
h = 1

Finally, let's find the value of yo by the equation of the second row and the third column:

c + 3f + i = 0
0 + 3 (1/2) + i = 0
3/2 + i = 0
i = 3/2


Vestibular exercises with feedback

1. (Cefet-MG) The matrix is ​​inverse of
It can be correctly stated that the difference (xy) is equal to:

a) -8
b) -2
c) 2
d) 6
e) 8

2. (UF Viçosa-MG) The matrices are:

Where x and y are real numbers and M is the inverse matrix of A. Therefore, the product xy is:

a) 3 / 2
b) 2/3
c) 1/2
d) 3/4
e) 1/4

3. (PUC-MG) The inverse matrix of the matrix is ​​equal to: