Question

If $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then the value of matrix $A =$

Answer 1

$\begin{bmatrix}1 & 1\\1 & 0\end{bmatrix}$

Answer 2

To find the matrix A, we need to isolate it from the given equation:

$\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Let $P = \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$ and $Q = \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix}$. Then the equation can be written as $PAQ = I$, where $I$ is the identity matrix. To solve for A, we need to find the inverses of P and Q and multiply them appropriately:

$A = P^{-1}Q^{-1}$

Step 1: Find $P^{-1}$

The determinant of $P$ is: $det(P) = (2)(2) - (1)(3) = 4 - 3 = 1$

Thus, the inverse of $P$ is: $P^{-1} = \begin{bmatrix} 2 & -1 \\ -3 & 2 \end{bmatrix}$

Step 2: Find $Q^{-1}$

The determinant of $Q$ is: $det(Q) = (-3)(-3) - (2)(5) = 9 - 10 = -1$

Thus, the inverse of $Q$ is: $Q^{-1} = \frac{1}{-1} \begin{bmatrix} -3 & -2 \\ -5 & -3 \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 5 & 3 \end{bmatrix}$

Step 3: Compute $A = P^{-1}Q^{-1}$

Now, multiply $P^{-1}$ and $Q^{-1}$: $A = \begin{bmatrix} 2 & -1 \\ -3 & 2 \end{bmatrix} \begin{bmatrix} 3 & 2 \\ 5 & 3 \end{bmatrix}$

Calculating the product: $A = \begin{bmatrix} (2*3 + (-1)*5) & (2*2 + (-1)*3) \\ (-3*3 + 2*5) & (-3*2 + 2*3) \end{bmatrix} = \begin{bmatrix} 6 - 5 & 4 - 3 \\ -9 + 10 & -6 + 6 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$

Therefore, $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$.