Unit 9 test: Systems and Matrices

Solve the system $4x - 3y = 5$ and $2x + y = 5$.

Solve the system.

A linear system reduces to the augmented matrix below. What can you conclude?

How many points do the circle $x^2 + y^2 = 9$ and the parabola $y = x^2 - 3$ have in common?

Let $A = \begin{bmatrix} 1 & 4 \\ -2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & -1 \\ 2 & 5 \end{bmatrix}$. What is the entry in row 1, column 2 of $2A - 3B$?

Let $A = \begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 0 \\ 1 & -1 \\ -1 & 3 \end{bmatrix}$. What is the entry in row 2, column 1 of $AB$?

$A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 3$ matrix. Which statement is true?

Which matrix is the inverse of $\begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}$?

Use the inverse matrix from the previous problem to solve $5x + 2y = 3$ and $2x + y = 1$.

Evaluate $\det A$ for $A = \begin{bmatrix} 3 & 0 & 2 \\ 1 & -1 & 4 \\ 2 & 5 & 1 \end{bmatrix}$.

Find all values of $k$ for which $\begin{bmatrix} k & 2 \\ 8 & k \end{bmatrix}$ has no inverse.

$A$ and $B$ are $2 \times 2$ matrices with $\det A = 3$ and $\det B = -2$. What is $\det(AB^{-1})$?

Find the area of the triangle with vertices $(-1, 2)$, $(3, 5)$ and $(4, -1)$.

Find $A$ and $B$ so that $\dfrac{3x + 4}{x^2 + x - 6} = \dfrac{A}{x + 3} + \dfrac{B}{x - 2}$. Give $(A, B)$.

The expression $\dfrac{4x^2 - x - 2}{x^2(x + 1)}$ can be written as $\dfrac{A}{x} + \dfrac{B}{x^2} + \dfrac{C}{x + 1}$. Find $C$.