Question-35

\(\displaystyle A,B\) and \(\displaystyle C\) are square matrices of order \(\displaystyle n\), \(\displaystyle 0\) is the zero matrix and \(\displaystyle I\) is an identity matrix of order \(\displaystyle n\). Compute the following product:

\[ \begin{bmatrix} I & 0\\ A & I \end{bmatrix}\begin{bmatrix} A & B\\ -C & 0 \end{bmatrix} \]

• \(\displaystyle \begin{bmatrix} A & B\\ C & I \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} B & A\\ A^{2} & BA \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} A & B\\ A^{2} -C & AB \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} A^{2} & B^{2}\\ AB & BA \end{bmatrix}\)

Answer

• \(\displaystyle \begin{bmatrix} A & B\\ C & I \end{bmatrix}\)
• \(\displaystyle \begin{bmatrix} B & A\\ A^{2} & BA \end{bmatrix}\)
• \(\displaystyle \begin{bmatrix} A & B\\ A^{2} -C & AB \end{bmatrix}\)
• \(\displaystyle \begin{bmatrix} A^{2} & B^{2}\\ AB & BA \end{bmatrix}\)

Solution

We can multiply the two block matrices as though the entries were scalars.