Respuesta :

The inverse of A is  [tex]A^{-1} = \left[\begin{array}{ccc}-10&9&-7\\14&-7&6\\-2&1&9\end{array}\right][/tex].

Matrix:

  • A matrix is a rectangular array of elements arranged in rows and columns.
  • It is denoted by a capital letter.

The inverse of a matrix:

  • The inverse of any matrix is that matrix, which when multiplied by the original matrix gives the identity matrix (all the diagonal elements = 1, and the rest are 0).
  • The inverse of a matrix A is denoted by [tex]A^{-1}[/tex] .

Step 1: Find the determinant of the given matrix. If it is not zero, then the matrix is invertible.

Let A = [tex]\left[\begin{array}{ccc}1&2&5\\3&5&9\\1&1&-2\end{array}\right][/tex] .

det A or |A|= 1(-10-9) - 2(-6-9) + 5(3-5) = -19+30-10 = 1.

Hence the matrix is invertible.

Step 2: Calculate the minors. Minors of an element of a matrix is the determinant obtained on removing that particular row and column.

[tex]a_{11}[/tex] = -10  [tex]a_{12}[/tex] = -14 [tex]a_{13}[/tex] = -2

[tex]a_{21}[/tex] = -9   [tex]a_{22}[/tex] = -7  [tex]a_{23}[/tex] = -1

[tex]a_{31}[/tex] = -7   [tex]a_{32}[/tex] = -6  [tex]a_{33}[/tex] = -1.

Step 3: Find the cofactors. Cofactors are calculated by multiplying the minor with -1 raised to the power of the sum of the row and column number of that element.

[tex]a_{11}[/tex] = -10  [tex]a_{12}[/tex] = +14 [tex]a_{13}[/tex] = -2

[tex]a_{21}[/tex] = +9   [tex]a_{22}[/tex] = -7  [tex]a_{23}[/tex] = +1

[tex]a_{31}[/tex] = -7   [tex]a_{32}[/tex] = +6  [tex]a_{33}[/tex] = -1.

Step 4: Calculate the adjoint matrix. The adjoint matrix is found by taking the transpose of the cofactor matrix.

Adjoint of A or adj A = [tex]\left[\begin{array}{ccc}-10&9&-7\\14&-7&6\\-2&1&9\end{array}\right][/tex]

Step 5: The inverse of a matrix A or [tex]A^{-1}[/tex] is calculated as:

[tex]A^{-1}[/tex] = [tex]\frac{1}{|A|}[/tex] x adj A

Since  [tex]\frac{1}{|A|}[/tex] = 1, here  [tex]A^{-1}[/tex] = adj A

So we get:

[tex]A^{-1} = \left[\begin{array}{ccc}-10&9&-7\\14&-7&6\\-2&1&9\end{array}\right][/tex]

Hence the inverse of A is  [tex]A^{-1} = \left[\begin{array}{ccc}-10&9&-7\\14&-7&6\\-2&1&9\end{array}\right][/tex].

The inverse of the given matrix is  [tex]\left[\begin{array}{ccc}-10&9&-7\\14&-7&6\\-2&1&9\end{array}\right][/tex].

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