Inverse Matrix and Simultaneous Equations

Question 1

The determinant of $\begin{pmatrix}3&2\1&4\end{pmatrix}$ is

Accepted Answer

10 — $\det=ad-bc=3(4)-2(1)=12-2=10$.

Question 2

A $2	imes 2$ matrix has an inverse only if its determinant is

Accepted Answer

not zero — The inverse divides by $ad-bc$, so it exists only when the determinant is non-zero.

Question 3

The inverse of $\begin{pmatrix}a&b\c&d\end{pmatrix}$ is

Accepted Answer

$\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$ — Swap $a,d$, negate $b,c$ and divide by the determinant $ad-bc$.

Question 4

The determinant of $\begin{pmatrix}5&2\10&4\end{pmatrix}$ is

Accepted Answer

0 — $\det=5(4)-2(10)=20-20=0$, so this matrix is singular.

Question 5

The inverse of $\begin{pmatrix}4&1\3&1\end{pmatrix}$ is

Accepted Answer

$\begin{pmatrix}1&-1\-3&4\end{pmatrix}$ — $\det=4(1)-1(3)=1$; inverse $=\dfrac{1}{1}\begin{pmatrix}1&-1\-3&4\end{pmatrix}$.

Question 6

To solve $A\mathbf{x}=\mathbf{b}$ using the inverse, $\mathbf{x}$ equals

Accepted Answer

$A^{-1}\mathbf{b}$ — Multiply $A\mathbf{x}=\mathbf{b}$ on the left by $A^{-1}$ to get $\mathbf{x}=A^{-1}\mathbf{b}$.

Question 7

For $\begin{pmatrix}2&1\3&2\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}4\7\end{pmatrix}$, the value of $x$ is

Accepted Answer

1 — $\det=2(2)-1(3)=1$; $A^{-1}=\begin{pmatrix}2&-1\-3&2\end{pmatrix}$; $x=2(4)-1(7)=8-7=1$.

Question 8

In the same system $\begin{pmatrix}2&1\3&2\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}4\7\end{pmatrix}$, the value of $y$ is

Accepted Answer

2 — $y=-3(4)+2(7)=-12+14=2$.

Question 9

The determinant of the identity matrix $\begin{pmatrix}1&0\0&1\end{pmatrix}$ is

Accepted Answer

1 — $\det=1(1)-0(0)=1$.

Question 10

For $\begin{pmatrix}1&2\3&4\end{pmatrix}$, the value of $ad-bc$ is

Accepted Answer

$-2$ — $ad-bc=1(4)-2(3)=4-6=-2$.