8.1 Operations of matrices (in class)

If [math]3\begin{bmatrix}1&7\\4&-5\end{bmatrix}-2\begin{bmatrix}7&-1\\-4&0\end{bmatrix}=\begin{bmatrix}a&b\\c&d\end{bmatrix}[/math] what are [math]a, b, c[/math], and [math]d[/math]?

If [math]\begin{bmatrix}-1&19\\2&3\end{bmatrix}\begin{bmatrix}2&0\\0&2\end{bmatrix}=\begin{bmatrix}a&b\\c&d\end{bmatrix}[/math] what are [math]a, b, c[/math], and [math]d[/math]?

Let [math]A=\begin{bmatrix}1&-3\\-2&7\end{bmatrix}[/math] and [math]B=\begin{bmatrix}2&7\\3&4\end{bmatrix}[/math]. If matrix [math]X=\begin{bmatrix}a&b\\c&d\end{bmatrix}[/math] is the solution to the equation [math]A-2X=B[/math], what is [math]X[/math]?

Which of the following matrix equations is equivalent to the equations system [math]\begin{cases}x+2y=7\\3x-4y=16\end{cases}[/math]?

Let [math]A_{20\times30}[/math] be a matrix. If [math]a_{ij}=(i-2j)^2[/math], how many entries in [math]A[/math] are [math]0[/math]?

If [math]\begin{bmatrix}2&1\\5&3\end{bmatrix}^{-1}=\begin{bmatrix}a&b\\c&d\end{bmatrix}[/math] what are [math]a, b, c[/math], and [math]d[/math]?

Let [math]A[/math] be a matrix satisfying [math]A^2-3A+2I=0[/math]. Express the following matrices in terms of [math]A[/math] and [math]I[/math]

(a) [math]A^3[/math]

(b) [math]A^4[/math]

If [math]A[/math] be an invertible matrix, which of the following matrices is equivalent to [math](AMA^{-1})^{n}[/math] where [math]n[/math] is a positive integer?

Consider three matrices [math]A_{3\times4}, B_{2\times4}[/math] and [math]C_{4\times2}[/math]. Which of the following matrices is/are undefined?

Let [math]A_{2\times2}[/math] be a matrix and [math]a_{11}=p, a_{12}=1[/math]. Suppose [math]A^2=0[/math]. Represent the following entries in terms of [math]p[/math].

(a) [math]a_{21}[/math]

(b) [math]a_{22}[/math]

Let [math]A=\begin{bmatrix}1&2\\-1&-3\end{bmatrix}[/math]

(a) Find constants [math]p, q[/math] such that [math]A^2=pA+qI[/math]

(b) Hence, represent [math]A^3[/math] in terms of [math]A[/math] and [math]I[/math].

Solve the matrix equation [math]\begin{bmatrix}2&0\\1&7\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}42\\-28\end{bmatrix}[/math]