1. Problem 10: Given ratios $x:y=2:3$ and $y:z=2:3$, find ratio $x:z$. Since $x:y=2:3$, we write $\frac{x}{y}=\frac{2}{3}$. Since $y:z=2:3$, we write $\frac{y}{z}=\frac{2}{3}$. We want $\frac{x}{z} = \frac{x}{y} \times \frac{y}{z} = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$. So $x:z=4:9$. Answer: D. 2. Problem 11: A matrix is singular if its determinant is... By definition, a matrix with determinant zero is singular. Answer: B (zero). 3. Problem 12: A matrix without an inverse is called... A matrix without an inverse is called singular. Answer: C (Singular). 4. Problem 13: Compute the additive identity of $A - BA - BA - B$. Given: $A=[2 \ 0 \ -3]$ $B=[-1 \ 0 \ -3]$ $C=\begin{bmatrix}5 \\ -2 \\ 4\end{bmatrix}^t$ Note these are row vectors for A and B and column vector for C. Additive identity means the zero matrix/vector matching the dimension. Regardless of the operation's values, the additive identity is the zero vector $[0\ 0\ 0]$. Answer: A. 5. Problem 14: Additive inverse of $A^t$. Transpose of $A$ is column vector $A^t=\begin{bmatrix}2 \\ 0 \\ -3\end{bmatrix}$. Additive inverse is negative of $A^t$: $-A^t=\begin{bmatrix}-2 \\ 0 \\ 3\end{bmatrix}$. Answer: A. 6. Problem 15: Order of matrix $C C C$. $C$ is $3 \times 1$. Matrix multiplication requires inner dimensions match. $C (3\times1) \times C (3\times1)$ is not defined. So $CCC$ is undefined unless clarified. But question likely means $C^t C C$ or similar. Given current info, $CCC$ dimension cannot be determined clearly; likely typo. Assuming $C$ is $3 \times 1$, repeated multiplication is not defined. But if it's vector dot product, result is scalar $1 \times 1$. Answer most fitting: A (1x1). 7. Problem 16: Add $B - C$ and $A - B$. Calculate $B - C$: $B = [-1\ 0\ -3]$, $C=[5\ -2\ 4]^t = [5\ -2\ 4]$. $B - C = [-1-5, 0-(-2), -3-4] = [-6, 2, -7]$. Calculate $A - B$: $A = [2\ 0\ -3]$, $B=[-1\ 0\ -3]$. $A - B = [2-(-1), 0-0, -3-(-3)] = [3, 0, 0]$. Now add $(B - C)+(A - B) = [-6+3, 2+0, -7+0] = [-3, 2, -7]$. None of the options match $[-3,2,-7]$, so answer is D (Cannot be determined). 8. Problem 17: Product matrix dimensions for $A_{3\times4} \times B_{4\times5}$. Matrix multiplication: result dimension is rows of first by columns of second. Result dimension = $3 \times 5$. Answer: C. Final answers: 10-D, 11-B, 12-C, 13-A, 14-A, 15-A, 16-D, 17-C.