8.1 Operations of matrices

If [math]\begin{bmatrix}-2&4\\18&6\end{bmatrix}+\begin{bmatrix}1&3\\6&4\end{bmatrix}=\begin{bmatrix}a&b\\c&d\end{bmatrix}[/math] what are [math]a, b, c[/math], and [math]d[/math]?

If [math]A=\begin{bmatrix}6&0&1\\2&3&7\\6&4&5\end{bmatrix}-\begin{bmatrix}9&0&2\\7&0&3\\1&8&-4\end{bmatrix}[/math], what are [math]a_{31}+a_{23}[/math]?

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]?

What is the maximum entry of the matrix [math]\displaystyle{\frac{1}{2}}\left(\begin{bmatrix}-5&3&9\\3&8&-1\end{bmatrix}+\begin{bmatrix}1&5&3\\7&-2&5\end{bmatrix}\right)[/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]?

What is the maximum entry of the matrix [math]\begin{bmatrix}7&3\\8&4\end{bmatrix}\begin{bmatrix}2&6&8&5\\6&0&0&1\end{bmatrix}[/math]?

What is the dimension of [math]\begin{bmatrix}1&0\\2&4\end{bmatrix}\begin{bmatrix}2&5\\6&1\\0&4\end{bmatrix}[/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]?

If the equations system [math]\begin{cases}x+z=10\\-x+2y+z=6\\3y-5z+7=0\end{cases}[/math] can be written as a matrix equation [math]AX=B[/math] where [math]X=\begin{bmatrix}x\\y\\z\end{bmatrix}[/math] what is the value of [math]a_{32}+b_{11}[/math]?

Consider two matrices [math]A=\begin{bmatrix}-2&a\\6&b\end{bmatrix}[/math] and [math]B=\begin{bmatrix}3&12\\c&0\end{bmatrix}[/math]. If [math]B=kA[/math] where [math]k[/math] is a constant, find [math]a, b[/math], and [math]c[/math].

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

If [math]A=\begin{bmatrix}1&8&9&4\\1&9&4&1\\1&9&4&9\\2&0&2&5\end{bmatrix}[/math] and [math]B=A^2[/math], what is [math]b_{31}[/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]

(Calculator) Let [math]A=\begin{bmatrix}-3&-2\\2&-1\\1&0\end{bmatrix}, B=\begin{bmatrix}-3&4&-5\\5&-4&3\end{bmatrix}[/math], and [math]C=\begin{bmatrix}7&-2&5\\1&2&1\end{bmatrix}[/math].

(a) What is the minimum entry of the matrix [math]2B-C[/math]?

(b) If [math]D=A(B+C)[/math], what is the value of [math]d_{13}+d_{21}[/math]?

(c) How many positive entries does the matrix [math]BA[/math] have?

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] and [math]q[/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].

Let [math]A[/math] be a [math]2018\times2018[/math] matrix and [math]a_{ij}=2i-j[/math]. Find the maximum entry of [math]A[/math]