Determine whether the following matrices are inverse matrices of one another \binom{5 \: 4}{3 \: 2} \\ \binom{2 \: \: \: -4}{ -3 \: \: 5} How to do this please help​

Question

Determine whether the following matrices are inverse matrices of one another


`\binom{5 \: 4}{3 \: 2} \\ \binom{2 \: \: \: -4}{ -3 \: \: 5}`
How to do this please help​

Answer 1

9514 1404 393

Answer:

they are not

Explanation:

The inverse matrix is the transpose of the cofactor matrix, divided by the determinant.

The cofactor matrix for a 2×2 matrix is ...

`\left[\begin{array}{cc}a_(22)&-a_(21)\\-a_(12)&a_(11)\end{array}\right]`

The transpose of this will have the off-diagonal terms swapped, so the inverse matrix is ...

`\displaystyle\frac{\left[\begin{array}{cc}a_(22)&-a_(12)\\-a_(21)&a_(11)\end{array}\right]}{a_(11)a_(22)-a_(21)a_(12)}`

We see that the second matrix is the transpose of the cofactor matrix, but the determinant is (5)(2)-(3)(4) = -2, so there has clearly been no division by the determinant. The actual inverse matrix of the first one shown is ...

`\left[\begin{array}{cc}-1&2\\1.5&-2.5\end{array}\right]`

_____

You can compute the matrix product to see if you get an identity matrix. Here, the upper left term in the product is ...

(5)(2) +(4)(-3) = -2 . . . . . not 1, so the product matrix is not an identity matrix

The matrices are not inverses.