1. a. Determine if $BA$ is defined and find its size. - Matrix $B$ is $4 \times 5$ and $A$ is $4 \times 5$. - For matrix multiplication $BA$, the number of columns of $B$ must equal the number of rows of $A$. - Here, $B$ has 5 columns, $A$ has 4 rows, so $5 \neq 4$. - Therefore, $BA$ is **not defined**. 1. b. Determine if $AB^T$ is defined and find its size. - $A$ is $4 \times 5$, $B$ is $4 \times 5$, so $B^T$ is $5 \times 4$. - For $AB^T$, columns of $A$ (5) must equal rows of $B^T$ (5), which is true. - The resulting matrix size is $4 \times 4$. 1. c. Determine if $AC + D$ is defined and find its size. - $A$ is $4 \times 5$, $C$ is $5 \times 2$, so $AC$ is $4 \times 2$. - $D$ is $4 \times 2$. - Since $AC$ and $D$ have the same size, $AC + D$ is defined and size is $4 \times 2$. 1. d. Determine if $E(AC)$ is defined and find its size. - $E$ is $5 \times 4$, $AC$ is $4 \times 2$. - For $E(AC)$, columns of $E$ (4) must equal rows of $AC$ (4), true. - Resulting size is $5 \times 2$. 2. a. Determine if $CD^T$ is defined and find its size. - $C$ is $5 \times 2$, $D$ is $4 \times 2$, so $D^T$ is $2 \times 4$. - For $CD^T$, columns of $C$ (2) must equal rows of $D^T$ (2), true. - Resulting size is $5 \times 4$. 2. b. Determine if $DC$ is defined and find its size. - $D$ is $4 \times 2$, $C$ is $5 \times 2$. - Columns of $D$ (2) must equal rows of $C$ (5), false. - $DC$ is **not defined**. 2. c. Determine if $BC - 3D$ is defined and find its size. - $B$ is $4 \times 5$, $C$ is $5 \times 2$, so $BC$ is $4 \times 2$. - $D$ is $4 \times 2$. - Both have same size, so $BC - 3D$ is defined and size is $4 \times 2$. 2. d. Determine if $D^T(BE)$ is defined and find its size. - $D$ is $4 \times 2$, so $D^T$ is $2 \times 4$. - $B$ is $4 \times 5$, $E$ is $5 \times 4$, so $BE$ is $4 \times 4$. - For $D^T(BE)$, columns of $D^T$ (4) must equal rows of $BE$ (4), true. - Resulting size is $2 \times 4$. 3. a. Compute $D + E$. - Both $D$ and $E$ are $3 \times 3$ matrices. - Add corresponding elements: $$D + E = \begin{bmatrix}1+6 & 5+1 & 2+3 \\ -1-1 & 0+1 & 1+2 \\ 3+4 & 2+1 & 4+3\end{bmatrix} = \begin{bmatrix}7 & 6 & 5 \\ -2 & 1 & 3 \\ 7 & 3 & 7\end{bmatrix}$$ 3. b. Compute $D - E$. - Subtract corresponding elements: $$D - E = \begin{bmatrix}1-6 & 5-1 & 2-3 \\ -1+1 & 0-1 & 1-2 \\ 3-4 & 2-1 & 4-3\end{bmatrix} = \begin{bmatrix}-5 & 4 & -1 \\ 0 & -1 & -1 \\ -1 & 1 & 1\end{bmatrix}$$ 3. c. Compute $5A$. - Multiply each element of $A$ by 5: $$5A = \begin{bmatrix}15 & 0 \\ -5 & 10 \\ 5 & 5\end{bmatrix}$$ 3. d. Compute $-7C$. - Multiply each element of $C$ by -7: $$-7C = \begin{bmatrix}-7 & -28 & -14 \\ -21 & -7 & -35\end{bmatrix}$$ 3. e. Compute $2B - C$. - Multiply $B$ by 2: $$2B = \begin{bmatrix}8 & -2 \\ 0 & 4\end{bmatrix}$$ - Subtract $C$ (note $B$ is $2 \times 2$, $C$ is $2 \times 3$, sizes differ, so $2B - C$ is **not defined**).