1. **State the problem:** We are given matrices with dimensions: - $A$ is $3 \times 2$ - $B$ is $2 \times 5$ - $C$ is $5 \times 4$ - $D$ is $4 \times 1$ - $E$ is $3 \times 3$ We need to find the dimensions of the following matrix expressions: - $B \cdot B^t$ - $E \cdot A$ - $-2D$ - $A^t + C^t$ 2. **Recall matrix multiplication and addition rules:** - For multiplication $X \cdot Y$, the number of columns of $X$ must equal the number of rows of $Y$. - The resulting matrix has dimensions (rows of $X$) $\times$ (columns of $Y$). - For addition $X + Y$, matrices must have the same dimensions. - Transpose $X^t$ swaps rows and columns of $X$. - Scalar multiplication $kX$ keeps the same dimensions as $X$. 3. **Calculate each expression:** - $B \cdot B^t$: - $B$ is $2 \times 5$, so $B^t$ is $5 \times 2$. - Multiplication is valid since $5 = 5$ (columns of $B$ and rows of $B^t$). - Resulting dimension: $2 \times 2$. - $E \cdot A$: - $E$ is $3 \times 3$, $A$ is $3 \times 2$. - Multiplication valid since $3 = 3$. - Resulting dimension: $3 \times 2$. - $-2D$: - Scalar multiplication does not change dimensions. - $D$ is $4 \times 1$, so $-2D$ is $4 \times 1$. - $A^t + C^t$: - $A$ is $3 \times 2$, so $A^t$ is $2 \times 3$. - $C$ is $5 \times 4$, so $C^t$ is $4 \times 5$. - Dimensions $2 \times 3$ and $4 \times 5$ are different, so addition is **not defined**. 4. **Summary of results:** - $B \cdot B^t$ is $2 \times 2$ - $E \cdot A$ is $3 \times 2$ - $-2D$ is $4 \times 1$ - $A^t + C^t$ is not defined due to dimension mismatch **Final answer:** $$\text{Dimensions: } B \cdot B^t = 2 \times 2, \quad E \cdot A = 3 \times 2, \quad -2D = 4 \times 1, \quad A^t + C^t \text{ undefined}$$