1. **State the problem:** We are given a matrix $A = \begin{bmatrix}-1 & 5 & 7 \\ 0 & -2 & 4\end{bmatrix}$, a vector $b = \begin{bmatrix}9 \\ 6\end{bmatrix}$, and a vector $x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$. We need to calculate four expressions involving the identity matrix $I$: (a) $AI$, (b) $IA$, (c) $Ix$, (d) $xI$ and indicate the dimension of each identity matrix. 2. **Recall properties:** The identity matrix $I_n$ of dimension $n \times n$ satisfies $AI_n = A$ and $I_mA = A$ for matrices $A$ of appropriate sizes. Also, for vectors, $Ix = x$ if $I$ is the identity matrix of dimension matching the vector. 3. **Dimensions:** - $A$ is a $2 \times 3$ matrix. - $b$ is a $2 \times 1$ vector. - $x$ is a $2 \times 1$ vector. 4. **Calculate each:** (a) $AI$: To multiply $A$ by an identity matrix on the right, $I$ must have dimension $3 \times 3$ to match $A$'s number of columns. Thus, $I_{3} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$. Then, $$AI_3 = A = \begin{bmatrix}-1 & 5 & 7 \\ 0 & -2 & 4\end{bmatrix}$$ (b) $IA$: To multiply identity matrix on the left of $A$, $I$ must be $2 \times 2$ since $A$ has 2 rows. Thus, $I_2 = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. However, $I_2$ ($2 \times 2$) times $A$ ($2 \times 3$) is not defined since inner dimensions mismatch: $(2 \times 2) \times (2 \times 3)$ is invalid. Therefore, $IA$ is undefined with the identity matrices defined as above. The correct multiplication $IA$ requires $I$ to be compatible. In practice, the product $IA$ implies $I = I_2$ acting on rows, so multiplication is done as $I_2 A$: this is valid. Hence, $$I_2 A = A = \begin{bmatrix}-1 & 5 & 7 \\ 0 & -2 & 4\end{bmatrix}$$ (c) $Ix$: $x$ is $2 \times 1$, so $I$ must be $2 \times 2$. $$I_2 = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, Ix = x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$$ (d) $xI$: $x$ is $2 \times 1$, multiplying on right by $I$ means $xI$ is $2 \times 1$ times $?$ which requires $I$ to be $1 \times 1$ identity matrix (a scalar 1). If $I$ is $1 \times 1$ identity matrix ($[1]$), $$xI = x \cdot 1 = x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$$ 5. **Summary:** - (a) $AI = A$ with $I$ of dimension $3 \times 3$. - (b) $IA = A$ with $I$ of dimension $2 \times 2$. - (c) $Ix = x$ with $I$ of dimension $2 \times 2$. - (d) $xI = x$ with $I$ of dimension $1 \times 1$.