Question

Does the matrix have an inverse? A = [3,-8,2,-5 ]

Answer 1

Yes, the matrix have an inverse and the inverse of the matrix is
`\left[\begin{array}{cc}-5&8\\-2&3\end{array}\right]`

Inverse of a matrix

The inverse of a matrix is another matrix that produces the multiplicative identity when multiplied by the given matrix. If we examine a matrix A, its inverse is denoted as A⁻¹. To get the inverse of a matrix, we would divide the adjugate of the provided matrix by the determinant of the given matrix.

Given that:

`A = \left[\begin{array}{cc}3 & -8 \\ 2 & -5 \end{array}\right]`

The determinant A of the 2 × 2 matrix is:

= (3 × -5) - (-8 × 2)

= - 15 + 16

= 1

This show that there is an inverse for this matrix since the determinant is not equal to zero. Thus, the inverse of the matrix can be computed as:

`Adj (A) = Adj \left[\begin{array}{cc}3 &-8\\ 2&5\\\end{array}\right]`

`\to \left[\begin{array}{cc}+(-5)&-(2)\\ -(-8)&(+3)\\\end{array}\right]^T`

`\to \left[\begin{array}{cc}-5&-2\\ 8&3\\\end{array}\right]`

`\to \left[\begin{array}{cc}-5&8\\ -2&3\\\end{array}\right]`

Now,
`A^(-1) = (1)/(|A|)* Adj (A)`

`=(1)/(1)* \left[\begin{array}{cc}-5&8\\-2&3\end{array}\right]`

`=\left[\begin{array}{cc}-5&8\\-2&3\end{array}\right]`