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Practice Test A

Give the order of the matrix and the value of the designated entry of the matrix.

$[\begin{array}{r} 3 & - 1 & - 3 & 0 \\ 2 & 4 & 1 & 7 \\ 5 & 0 & 3 & 2 \\ 8 & 10 & 9 & 6 \\ - 4 & 0 & 0 & 3 \end{array}] a_{43}$ $[\begin{array}{r} 3 & - 1 & - 3 & 0 \\ 2 & 4 & 1 & 7 \\ 5 & 0 & 3 & 2 \\ 8 & 10 & 9 & 6 \\ - 4 & 0 & 0 & 3 \end{array}] a_{43}$
Write an augmented matrix for the system of equations.

${\begin{array}{rclrrl} 7 x & - & 3 y & + & 9 z & = & 5 \\ - 2 x & + & 4 y & + & 3 z & = & - 12 \\ 8 x & - & 5 y & + & z & = & - 9 \end{array}$ ${\begin{array}{rclrrl} 7 x & - & 3 y & + & 9 z & = & 5 \\ - 2 x & + & 4 y & + & 3 z & = & - 12 \\ 8 x & - & 5 y & + & z & = & - 9 \end{array}$
Write the system of linear equations represented by the augmented matrix

$[\begin{array}{r} 4 & 0 & - 1 & - 3 \\ 1 & 3 & 0 & 9 \\ 2 & 7 & 5 & 8 \end{array}] .$ $[\begin{array}{r} 4 & 0 & - 1 & - 3 \\ 1 & 3 & 0 & 9 \\ 2 & 7 & 5 & 8 \end{array}] .$

Use x, y, and z for the variables.
Find values for the variables so that the matrices

$[\begin{array}{r} x & - & 5 & y & + & 3 \\ z & 9 \end{array}] = [\begin{array}{r} 5 & 13 \\ 0 & 9 \end{array}]$ $[\begin{array}{r} x & - & 5 & y & + & 3 \\ z & 9 \end{array}] = [\begin{array}{r} 5 & 13 \\ 0 & 9 \end{array}]$

are equal.

In Problems 5 and 6, solve the system of equations by using matrices.

${\begin{array}{rcrrrl} x & + & 2 y & + & z & = & 6 \\ x & + & y & - & z & = & 7 \\ 2 x & - & y & + & 2 z & = & - 3 \end{array}$ ${\begin{array}{rcrrrl} x & + & 2 y & + & z & = & 6 \\ x & + & y & - & z & = & 7 \\ 2 x & - & y & + & 2 z & = & - 3 \end{array}$
${\begin{array}{rcrrrl} 2 x & + & y & - & 4 z & = & 6 \\ - x & + & 3 y & - & z & = & - 2 \\ 2 x & - & 6 y & + & 2 z & = & 4 \end{array}$ ${\begin{array}{rcrrrl} 2 x & + & y & - & 4 z & = & 6 \\ - x & + & 3 y & - & z & = & - 2 \\ 2 x & - & 6 y & + & 2 z & = & 4 \end{array}$
Let $A = [\begin{array}{r} 5 & - 2 \\ 4 & 0 \\ 7 & 6 \end{array}] and B = [\begin{array}{r} - 3 & - 1 \\ 0 & - 8 \\ - 4 & 6 \end{array}] .$ $A = [\begin{array}{r} 5 & - 2 \\ 4 & 0 \\ 7 & 6 \end{array}] and B = [\begin{array}{r} - 3 & - 1 \\ 0 & - 8 \\ - 4 & 6 \end{array}] .$ Find $A - B .$ $A - B .$

In Problems 8 and 9, find the product AB or state that AB is not defined.

$A = [\begin{array}{r} 3 & - 7 & 2 \end{array}], B = [\begin{array}{r} 0 \\ 1 \\ 4 \end{array}]$ $A = [\begin{array}{r} 3 & - 7 & 2 \end{array}], B = [\begin{array}{r} 0 \\ 1 \\ 4 \end{array}]$
$A = [\begin{array}{r} - 4 & 1 & 2 \\ - 9 & 0 & 3 \end{array}], B = [\begin{array}{r} 2 & 8 \\ 7 & - 5 \end{array}]$ $A = [\begin{array}{r} - 4 & 1 & 2 \\ - 9 & 0 & 3 \end{array}], B = [\begin{array}{r} 2 & 8 \\ 7 & - 5 \end{array}]$

For Problems 10–13, let

A = [\begin{array}{r} - 3 & 1 & 0 \\ 5 & 7 & 2 \end{array}], B = [\begin{array}{r} - 1 & 4 \\ 8 & 2 \end{array}], and C = [\begin{array}{r} 1 & 5 \\ 0 & 4 \end{array}] .

$A = [\begin{array}{r} - 3 & 1 & 0 \\ 5 & 7 & 2 \end{array}], B = [\begin{array}{r} - 1 & 4 \\ 8 & 2 \end{array}], and C = [\begin{array}{r} 1 & 5 \\ 0 & 4 \end{array}] .$

Find 2A.
Find $A + B A .$ $A + B A .$
Find $C^{2} .$ $C^{2} .$
Find $C^{- 1} .$ $C^{- 1} .$
Find the inverse of the matrix

$A = [\begin{array}{r} 2 & 1 & 3 \\ 1 & 2 & - 1 \\ 3 & 1 & 5 \end{array}] .$ $A = [\begin{array}{r} 2 & 1 & 3 \\ 1 & 2 & - 1 \\ 3 & 1 & 5 \end{array}] .$
Write the linear system

${\begin{array}{rcrl} 5 x & + & 2 y & = & 32 \\ 3 x & + & y & = & 18 \end{array}$ ${\begin{array}{rcrl} 5 x & + & 2 y & = & 32 \\ 3 x & + & y & = & 18 \end{array}$

as a matrix equation in the form $A X = B,$ $A X = B,$ where A is the coefficient matrix and B is the constant matrix.
Write the matrix equation

$[\begin{array}{r} 12 & - 3 \\ - 2 & 7 \end{array}] [\begin{array}{r} x \\ y \end{array}] = [\begin{array}{r} 5 \\ - 9 \end{array}]$ $[\begin{array}{r} 12 & - 3 \\ - 2 & 7 \end{array}] [\begin{array}{r} x \\ y \end{array}] = [\begin{array}{r} 5 \\ - 9 \end{array}]$

as a system of linear equations without matrices.
Solve the matrix equation $A - 5 X = 2 B$ $A - 5 X = 2 B$ for X, where

$A = [\begin{array}{r} 1 & 5 & - 2 \\ 4 & - 2 & 7 \end{array}] and B = [\begin{array}{r} 2 & 5 & - 11 \\ 18 & 8 & 11 \end{array}] .$ $A = [\begin{array}{r} 1 & 5 & - 2 \\ 4 & - 2 & 7 \end{array}] and B = [\begin{array}{r} 2 & 5 & - 11 \\ 18 & 8 & 11 \end{array}] .$

In Problems 18 and 19, evaluate the determinant.

$| \begin{array}{r} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{2} & \frac{3}{4} \end{array} |$ $| \begin{array}{r} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{2} & \frac{3}{4} \end{array} |$
$| \begin{array}{r} 1 & 3 & 5 \\ 2 & 0 & 10 \\ - 3 & 1 & - 15 \end{array} |$ $| \begin{array}{r} 1 & 3 & 5 \\ 2 & 0 & 10 \\ - 3 & 1 & - 15 \end{array} |$
Use Cramer’s Rule to write the solution of the system

${\begin{array}{rcrrrl} 2 x & - & y & + & z & = & 3 \\ x & + & y & + & z & = & 6 \\ 4 x & + & 3 y & - & 2 z & = & 4 \end{array}$ ${\begin{array}{rcrrrl} 2 x & - & y & + & z & = & 3 \\ x & + & y & + & z & = & 6 \\ 4 x & + & 3 y & - & 2 z & = & 4 \end{array}$

in the determinant form. (You do not need to find the solution.)

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Table of Contents for Practice Test A

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Table of Contents for
Practice Test A