1. **Problem Statement:** We are given matrices $A$, $B$, $C$, $D$ and vectors $u$, $v$, $w$, $E$. We need to evaluate expressions involving addition and scalar multiplication of these matrices and vectors, and identify which operations are defined or not based on matrix dimensions. 2. **Important Rules:** - Matrices can be added or subtracted only if they have the same dimensions. - Scalar multiplication multiplies every element of the matrix or vector by the scalar. - Vector addition and subtraction require vectors of the same size. - If dimensions do not match, the operation is not defined. 3. **Given Matrices and Vectors:** $$ A = \begin{bmatrix}0 & 2 & 4 \\ 6 & 5 & 5 \\ 1 & 0 & -3\end{bmatrix},\quad B = \begin{bmatrix}0 & 5 & 2 \\ 5 & 3 & 4 \\ -2 & 4 & -2\end{bmatrix} $$ $$ C = \begin{bmatrix}5 & 2 \\ -2 & 4 \\ 1 & 0\end{bmatrix},\quad D = \begin{bmatrix}-4 & 1 \\ 5 & 0 \\ 2 & -1\end{bmatrix} $$ $$ E = \begin{bmatrix}3 & 4 \\ 3 & -1\end{bmatrix} $$ $$ u = \begin{bmatrix}1.5 \\ 0 \\ -3.0\end{bmatrix},\quad v = \begin{bmatrix}-1 \\ 3 \\ 2\end{bmatrix},\quad w = \begin{bmatrix}-5 \\ -30 \\ 10\end{bmatrix} $$ 4. **Step-by-step Solutions:** **8. Expressions:** - $2A + 4B$: Both $A$ and $B$ are $3\times3$, so addition is defined. Calculate $2A$ and $4B$ by multiplying each element by the scalar, then add element-wise. - $4B + 2A$: Same as above, addition is commutative. - $0A + B$: $0A$ is a zero matrix of same size as $A$, so $0A + B = B$. - $0.4B - 4.2A$: Scalar multiply and subtract element-wise. **9. Expressions:** - $3A$: Multiply each element of $A$ by 3. - $0.5B$: Multiply each element of $B$ by 0.5. - $3A + 0.5B$: Add the two scaled matrices element-wise. - $3A + 0.5B + C$: $C$ is $3\times2$, $A$ and $B$ are $3\times3$, so addition with $C$ is not defined due to dimension mismatch. **10. Expressions:** - $(4 - 3)A = 1A = A$. - $4(3A) = 12A$. - $14B - 3B = (14 - 3)B = 11B$. - $11B$: scalar multiplication. **11. Expressions:** - $8C + 10D$: Both $C$ and $D$ are $3\times2$, addition defined. - $2(5D + 4C)$: Compute $5D + 4C$ first, then multiply by 2. - $0.6C - 0.6D$: Scalar multiply and subtract. - $0.6(C - D)$: Same as above, distribute scalar. **12. Expressions:** - $(C - D) + E$: $C$ and $D$ are $3\times2$, $E$ is $2\times2$, addition with $E$ not defined. - $(D + E) + C$: $D + E$ not defined (different sizes), so whole expression not defined. - $0(C - E) - 4D$: $C - E$ not defined, so expression not defined. - $A - 0C$: $A$ is $3\times3$, $C$ is $3\times2$, subtraction not defined. **13. Expressions:** - $(2 - 7)C = -5C$. - $2(7C) = 14C$. - $-D + 0E$: $D$ is $3\times2$, $E$ is $2\times2$, addition not defined. - $E - D + C + u$: $E$, $D$, $C$ have incompatible sizes, and $u$ is a vector, so expression not defined. **14. Expressions:** - $(5u - 5v)$: $u$ and $v$ are vectors of size 3, subtraction defined. - $\frac{1}{2} w$: scalar multiply vector $w$. - $-20(u - v) + 2w$: compute $u - v$, scalar multiply by -20, add $2w$. - $E - (u + v)$: $E$ is $2\times2$ matrix, $u+v$ is vector, addition not defined. - $10(u - v) - w$: scalar multiply and subtract vectors. **15. Expressions:** - $(u + v) - w$: vector addition and subtraction defined. - $u - (v - w)$: vector subtraction defined. - $C - 0w$: $C$ is matrix, $w$ is vector, subtraction not defined. - $0E - u - v$: $0E$ is matrix, $u$, $v$ vectors, subtraction not defined. **16. Expressions:** - $15v - 3w - 0u$: vector operations defined. - $-3w + 15v$: vector operations defined. - $D - u - 3C$: $D$ and $C$ are matrices, $u$ vector, subtraction not defined. - $8.5w - 11.1u - 0.4v$: vector operations defined. **Summary:** - Matrix addition/subtraction requires same dimensions. - Vector addition/subtraction requires same size. - Scalar multiplication always defined. - Expressions mixing incompatible sizes are not defined. This completes the evaluation and explanation of the expressions.