Find the matrix X for which: [(3, 2), (7, 5)]X [(-1, 1), (-2, 1)]=[(2, -1), (0, 4)].

Question

Find the matrix X for which:

\(\begin{bmatrix}3&2\\7&5\end{bmatrix}X\begin{bmatrix}-1&1\\-2&1\end{bmatrix}\)\(=\begin{bmatrix}2&-1\\0&4\end{bmatrix}.\)

Answer 1

Let A = \(\begin{bmatrix}3&2\\7&5\end{bmatrix}\), B = \(\begin{bmatrix}-1&1\\-2&1\end{bmatrix}\), C = \(\begin{bmatrix}2&-1\\0&4\end{bmatrix}\)

Then given matrix equation becomes

A x B = C

⇒ A^-1 (A x B) B^-1 = A^-1 C B^-1

⇒ (A^-1A) x (BB^-1) = A^-1CB^-1

⇒ x = A^-1CB^-1

Now, |A| = 15 - 14 = 1

|B| = -1 + 2 = 1

adj (A) = \(\begin{bmatrix}5&-2\\-7&3\end{bmatrix}\)

 adj (B) = \(\begin{bmatrix}1&-1\\2&-1\end{bmatrix}\)

 ∵ A^-1 = \(\frac{adj(A)}{|A|}\) = \(\begin{bmatrix}5&-2\\-7&3\end{bmatrix}\)

 B^-1 = \(\frac{adj(B)}{|B|}\) = \(\begin{bmatrix}1&-1\\2&-1\end{bmatrix}\)

Now, 

x = A^-1cB^-1 

 = \(\begin{bmatrix}5&-2\\-7&3\end{bmatrix}\) \(\begin{bmatrix}2&-1\\0&4\end{bmatrix}\) \(\begin{bmatrix}1&-1\\2&-1\end{bmatrix}\) 

 = \(\begin{bmatrix}10&-13\\-14&-19\end{bmatrix}\) \(\begin{bmatrix}1&-1\\2&-1\end{bmatrix}\) 

 = \(\begin{bmatrix}-16&3\\24&-5\end{bmatrix}\)

Answer 2

Let  \(A=\begin{bmatrix}3&2\\7&5\end{bmatrix}\)\(B=\begin{bmatrix}-1&1\\-2&1\end{bmatrix}\)\(C=\begin{bmatrix}2&-1\\0&4\end{bmatrix}.\)  

Then The given equations becomes as

AXB = C

= X = A^–1CB^–1

|A| = 35 – 14 = 21

|B| = – 1 + 2 = 1

Comments

Wrong answer