# The 2 times 2 matrices A and B below are related to matrix C by the equation: C=3A-2B. Which of the following is matrix C. A=begin{bmatrix}3 & 5 -2 & 1 end{bmatrix} B=begin{bmatrix}-4 & 5 2 & 1 end{bmatrix} begin{bmatrix}-1 & 5 2 & 1 end{bmatrix} begin{bmatrix}-18 & 5 10 & 1 end{bmatrix} begin{bmatrix}18 & -5 -10 & -1 end{bmatrix} begin{bmatrix}1 & -5 -2 & -1 end{bmatrix}

Question
Matrices
The 2 \times 2 matrices A and B below are related to matrix C by the equation: C=3A-2B. Which of the following is matrix C.
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}$$
$$\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}$$
$$\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}$$

2020-12-26
Step 1
We have to find matrix C by the equation C=3A−2B where matrices are given as:
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
We know the operations of matrices,
If we multiply by any scalar to the matrix then it get multiplied in each elements example:
$$2\begin{bmatrix}a & b \\c & d \end{bmatrix}=\begin{bmatrix}2a & 2b \\2c & 2d \end{bmatrix}$$
$$\begin{bmatrix}a & b \\c & d \end{bmatrix}+\begin{bmatrix}x & y \\z & w \end{bmatrix}=\begin{bmatrix}a+x & b+y \\c+z & d+w \end{bmatrix}$$
Step 2
Applying above rule for the given condition, we get
$$C=3A-2B$$
$$=3\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix}-2\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}3 \times 3 & 3\times5 \\3\times (-2) & 3\times 1 \end{bmatrix}-\begin{bmatrix}2\times (-4) & 2\times 5 \\2\times 2 & 2\times 1 \end{bmatrix}$$
$$=\begin{bmatrix}9 & 15 \\-6 & 3 \end{bmatrix}-\begin{bmatrix}-8 & 10 \\4 & 2 \end{bmatrix}$$
$$=\begin{bmatrix}9-(-8) & 15-10 \\-6-4 & 3-2 \end{bmatrix}$$
$$=\begin{bmatrix}9+8 & 5 \\-10 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}17 & 5 \\-10 & 1 \end{bmatrix}$$
Hence, value of C is $$\begin{bmatrix}17 & 5 \\-10 & 1 \end{bmatrix}$$
Note:
There is no suitable option for the given conditions.

### Relevant Questions

Find the matrices:
a)A + B
b) A - B
c) -4A
d)3A + 2B
$$A=\begin{bmatrix}4 & 1 \\ 3 & 2 \end{bmatrix} ,B=\begin{bmatrix}5 & 9 \\ 0 & 7 \end{bmatrix}$$
Find the following matrices: a) A + B.
(b) A - B.
(c) -4A.
(d) 3A + 2B.
$$A=\begin{bmatrix}6 & 2 & -3 \end{bmatrix} , B=\begin{bmatrix}4 & -2 & 3 \end{bmatrix}$$
Given the matrices
$$A=\begin{bmatrix}-1 & 3 \\2 & -1 \\ 3&1 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -2 \\1 & 3 \\ 4 & -3 \end{bmatrix}$$ find the $$3 \times 2$$ matrix X that is a solution of the equation. 8X+A=B
Refer to the following matrices.
$$A=\begin{bmatrix}2 & -3&7&-4 \\-11 & 2&6&7 \\6 & 0&2&7 \\5 & 1&5&-8 \end{bmatrix} B=\begin{bmatrix}3 & -1&2 \\0 & 1&4 \\3 & 2&1 \\-1 & 0&8 \end{bmatrix} , C=\begin{bmatrix}1& 0&3 &4&5 \end{bmatrix} , D =\begin{bmatrix}1\\ 3\\-2 \\0 \end{bmatrix}$$
Identify the row matrix. Matrix C is a row matrix.
Given the matrices
$$A=\begin{bmatrix}1& -1&2 \\3&4&5\\0&1&-1 \end{bmatrix} , B=\begin{bmatrix}0&2&1 \\3&0&5\\7&-6&0 \end{bmatrix} \text{ and } C=\begin{bmatrix}0&0&2 \\3&1&0\\0&-2&4 \end{bmatrix}$$
Determine the following
i)2A-B+2C ii)A+B+C iii)4C-2B+3A iv)$$(A \times B)-C$$
Given the matrices
$$A=\begin{bmatrix}5 & 3 \\ -3 & -1 \\ -2 & -5 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -2 \\ 1 & 3 \\ 4 & -3 \end{bmatrix}$$
find the 3x2 matrix X that is a solution of the equation. 2X-A=X+B
X=?
Matrices C and D are shown below
C=\begin{bmatrix}2&1&0 \\0&3&4\\0&2&1 \end{bmatrix},D=\begin{bmatrix}a & b&-0.4 \\0&-0.2&0.8\\0&0.4&-0.6 \end{bmatrix}
What values of a and b will make the equation CD=I true?
a)a=0.5 , b=0.1
b)a=0.1 , b=0.5
c)a=-0.5 , b=-0.1
Consider the matrices
$$A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}$$
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $$AX^T=BC^3$$
(ii) $$A^{-1}(X-T)^T=(B^{-1})^T$$
(iii) $$XF=F^{-1}-D^T$$
a) $$\begin{bmatrix}0 & 3 \\1 & 0 \end{bmatrix}$$
b) $$\begin{bmatrix}2 & 0 \\0 & 1 \end{bmatrix}$$
c) $$\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
d) $$\begin{bmatrix}2 & 0 \\0 & 2 \end{bmatrix}$$
$$\begin{bmatrix}2 & -1&4 \\g & 0&3\\2&h&0 \end{bmatrix} \times \begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}=\begin{bmatrix}i & 24 \\-16&-4\\4&e \end{bmatrix}$$