Question

Determine whether the following matrices are inverse matrices of one another

Answer 1

9514 1404 393

Answer:

  they are not

Step-by-step 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.