Matrix Multiplication

Each entry of \mathbf{A}\mathbf{B} is a row of \mathbf{A} times a column of \mathbf{B}, with the products added.

1. Top left: row 1 of \mathbf{A} \times column 1 of \mathbf{B}: 3(1)+(-2)(-3)=3+6=9
2. Top right: row 1 \times column 2: 3(6)+(-2)(2)=18-4=14
3. Bottom left: row 2 \times column 1: 5(1)+4(-3)=5-12=-7
4. Bottom right: row 2 \times column 2: 5(6)+4(2)=30+8=38
5. Order matters: \mathbf{B}\mathbf{A} is a different matrix.
6. Answer: \begin{pmatrix}9&14\\-7&38\end{pmatrix}

Write down \mathbf{I}, then use a row of \mathbf{A} times the column of \mathbf{B} that contains k.

1. \mathbf{I}=\begin{pmatrix}1&0\\0&1\end{pmatrix} (1s on the leading diagonal, 0s elsewhere).
2. Top right of \mathbf{A}\mathbf{B}: row 1 \times column 2: 2(-1)+1(k)=k-2
3. This must equal the top right of \mathbf{I}, which is 0: k-2=0
4. Check with the bottom right: 5(-1)+3k=-5+6=1 ✓
5. Answer: k=2

Multiply \mathbf{A} by itself row-by-column, then compare entries: the diagonal entries give two possibilities each, and an off-diagonal entry tells you which pair works.

Part (a)

1. \mathbf{A}^2=\mathbf{A}\mathbf{A}=\begin{pmatrix}a&2\\1&b\end{pmatrix}\begin{pmatrix}a&2\\1&b\end{pmatrix}=\begin{pmatrix}a^2+2&2a+2b\\a+b&2+b^2\end{pmatrix}
2. (Not \begin{pmatrix}a^2&4\\1&b^2\end{pmatrix}: you can't square each entry.)
3. Top left: a^2+2=11, so a^2=9 and a=3 or a=-3
4. Bottom right: 2+b^2=6, so b^2=4 and b=2 or b=-2
5. Bottom left: a+b=1 (top right 2a+2b=2 says the same).
6. The only pair that adds to 1 is a=3, b=-2
7. Answer: a=3

Part (b)

1. From part (a), a+b=1 with a=3, so b=-2 (and b^2=4 ✓).
2. Answer: b=-2

Part (c)

1. \mathbf{A}=\begin{pmatrix}3&2\\1&-2\end{pmatrix}
2. \mathbf{A}^2-\mathbf{A}=\begin{pmatrix}11-3&2-2\\1-1&6-(-2)\end{pmatrix}=\begin{pmatrix}8&0\\0&8\end{pmatrix}
3. \begin{pmatrix}8&0\\0&8\end{pmatrix}=8\begin{pmatrix}1&0\\0&1\end{pmatrix}=8\mathbf{I}
4. Answer: k=8