Find the dimensions of the given matrix and identify the given entry. $$ A=\left[\begin{array}{llll} 1 & 5 & 0 & \frac{1}{4} \end{array}\right] ; a_{13} $$

Solve the matrix equation \(A(B+C X)=D\) for \(X\). (You may assume that all the matrices are square and invertible.)

Translate the given matrix equations into svstems of linear equations. $$ \left[\begin{array}{rrr} 1 & -1 & 4 \\ -\frac{1}{3} & -3 & \frac{1}{3} \\ 3 & 0 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} -3 \\ -1 \\ 2 \end{array}\right] $$

Translate the given matrix equations into svstems of linear equations. $$ \left[\begin{array}{lrll} 0 & 1 & 6 & 1 \\ 1 & -5 & 0 & 0 \end{array}\right]\left[\begin{array}{r} x \\ y \\ z \\ w \end{array}\right]=\left[\begin{array}{r} -2 \\ 9 \end{array}\right] $$

Define the naive product \(A \square B\) of two \(m \times n\) matrices \(A\) and \(B\) by $$ (A \square B)_{i j}=A_{i j} B_{i j} $$ (This is how someone who has never seen matrix multiplication before might think to multiply matrices.) Referring to Example 1 in this section, compute and comment on the meaning of \(P \square\left(Q^{T}\right.\).)

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\\1 & 0 \\\\-1 & 2 \end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\\0 & 0.5 \\\\-1 & 3\end{array}\right], \text { and } \\\&C=\left[\begin{array}{rr}1 & -1 \\\1 & 1 \\\\-1 & -1\end{array}\right].\end{aligned}$$ $$ A^{T}+3 C^{T} $$