# find which of the given matrices are nonsingular. a) begin{bmatrix}1 & 2 &-3 -1 & 2&3 0 &8&0 end{bmatrix} b)begin{bmatrix}1 & 2 &-3 -1 & 2&3 0 &1&1 end{bmatrix} c) begin{bmatrix}1 & 1 &2 -1 & 3&4 -5 &7&8 end{bmatrix} d) begin{bmatrix}1 & 1 &4&-1 1 & 2&3&2 -1 &3&2&1-2&6&12&-4 end{bmatrix}

Question
Matrices
find which of the given matrices are nonsingular.
a) $$\begin{bmatrix}1 & 2 &-3 \\-1 & 2&3 \\ 0 &8&0 \end{bmatrix}$$
b)$$\begin{bmatrix}1 & 2 &-3 \\-1 & 2&3 \\ 0 &1&1 \end{bmatrix}$$
c) $$\begin{bmatrix}1 & 1 &2 \\-1 & 3&4 \\ -5 &7&8 \end{bmatrix}$$
d) $$\begin{bmatrix}1 & 1 &4&-1 \\1 & 2&3&2 \\ -1 &3&2&1\\-2&6&12&-4 \end{bmatrix}$$

2021-02-10
Step 1
Since you have posted a multiple sub-parts problems ,we solve first 3 sub-parts for you .To get the remaining sub-parts solved please repost the complete question ,and mention sub-parts to be solved.
We have given some matrices and we have to determine which matrices are non-singular among them.
Step 2 First note that a matrix A is said to be nonsingular if $$det(A) \neq 0$$.We will use this to determine the non singularity of the matrices .
a) we have $$A=\begin{bmatrix}1 & 2 &-3 \\-1 & 2&3 \\ 0 &8&0 \end{bmatrix}$$
Now $$det(A)=1 \times (0-24)-2 \times (0-0)-3 \times (-8-0)=-24+24=0$$
Since determinant is 0 so the matrix is singular ,not non-singular.
b) $$B=\begin{bmatrix}1 & 2 &-3 \\-1 & 2&3 \\ 0 &1&1 \end{bmatrix}$$
Now $$det(B)=1 \times (2-3)- 2 \times (-1-0)-3 \times (-1-0)=-1+2+3=4 \neq 0$$
Sine the determinant is non-zero so the matrix is non-singular .
c) $$C=\begin{bmatrix}1 & 1&2 \\-1 & 3&4 \\ -5 &7&8 \end{bmatrix}$$
Now, $$det(C)=1 \times (24-28)-1 \times (-8+20)+2 \times (-7+15)=0$$
Since determinant is 0 so the matrix is singular ,not non-singular.

### Relevant Questions

compute the indicated matrices (if possible). B - C
Let
$$A=\begin{bmatrix}3 & 0 \\-1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2&1 \\0 & 2&3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\3 & 4\\5&6 \end{bmatrix}, D=\begin{bmatrix}0 & -3 \\-2 & 1 \end{bmatrix},E=\begin{bmatrix}4 & 2 \end{bmatrix},F=\begin{bmatrix}-1 \\2 \end{bmatrix}$$
compute the indicated matrices . FE
$$A=\begin{bmatrix}3 & 0 \\-1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2&1 \\0 & 2&3 \end{bmatrix} , C=\begin{bmatrix}1& 2 \\3 & 4\\5&6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\-2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} ,F=\begin{bmatrix}-1 \\2 \end{bmatrix}$$
Use the graphing calculator to solve if possible
A=\begin{bmatrix}1 & 0&5 \\1 & -5&7\\0&3&-4 \end{bmatrix}\\ B=\begin{bmatrix}3 & -5&3 \\2&3&1\\4&1&-3\end{bmatrix}\\ C=\begin{bmatrix}5 & 2&3 \\2& -1&0 \end{bmatrix}\\ D=\begin{bmatrix}5 \\-3\\4 \end{bmatrix}
Find the value in row 2 column 3 of AB-3B
$$A=\begin{bmatrix}2& 1&1 \\-1 & -1&4 \end{bmatrix} B=\begin{bmatrix}0& 2 \\-4 & 1\\2 & -3 \end{bmatrix} C=\begin{bmatrix}6& -1 \\3 & 0\\-2 & 5 \end{bmatrix} D=\begin{bmatrix}2& -3&4 \\-3 & 1&-2 \end{bmatrix}$$
a)$$A-3D$$
b)$$B+\frac{1}{2}$$
c) $$C+ \frac{1}{2}B$$
(a),(b),(c) need to be solved
If $$A=\begin{bmatrix}1 & 1 \\3 & 4 \end{bmatrix} , B=\begin{bmatrix}2 \\1 \end{bmatrix} ,C=\begin{bmatrix}-7 & 1 \\0 & 4 \end{bmatrix},D=\begin{bmatrix}3 & 2 & 1 \end{bmatrix} \text{ and } E=\begin{bmatrix}2 & 3&4 \\1 & 2&-1 \end{bmatrix}$$
Find , if possible,
a) A+B , C-A and D-E b)AB, BA , CA , AC , DA , DB , BD , EB , BE and AE c) 7C , -3D and KE
Find the matrices:
a)A + B
b) A - B
c) -4A
d)3A + 2B
$$A=\begin{bmatrix}4 & 1 \\ 3 & 2 \end{bmatrix} ,B=\begin{bmatrix}5 & 9 \\ 0 & 7 \end{bmatrix}$$
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.
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}$$
$$\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}$$
$$\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}$$
Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
$$A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}$$
If the operation is not possible , write NOT POSSIBLE and be able to explain why
a)A+B
b)B+C
c)2A
Given the matrices $$A=\begin{bmatrix}-4 & -3&2 \\ 0 & 2 &-2 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -1&5 \\ 2 & -1 &2 \end{bmatrix}$$ A+B=?
$$A=\begin{bmatrix}5 & 3 \\ -3 & -1 \\ -2 & -5 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -2 \\ 1 & 3 \\ 4 & -3 \end{bmatrix}$$