Chapter 8

Fill in the blank. A system of equations that is in _____ form has a "stair-step" pattern with leading coefficients of \(1 .\)

Fill in the blank. _______ is a method for using determinants to solve a system of linear equations.

Fill in the blank. Both \(\operatorname{det}(A)\) and \(|A|\) represent the ______ of the matrix \(A\)

fill in the blank(s). If there exists an \(n \times n\) matrix \(A^{-1}\) such that \(A A^{-1}=I_{n}=A^{-1} A,\) then \(A^{-1}\) is called the ________of \(A\).

Fill in the blank. A rectangular array of real numbers that can be used to solve a system of linear equations is called a _________________ .

The first step in solving a system of equations by the _____ of _____ is to obtain coefficients for \(x\) (or \(y\) ) that differ only in sign.

Two matrices are _____ when they have the same dimension and all of their corresponding entries are equal.

Fill in the blank. A solution of a system of three linear equations in three unknowns can be written as an ____, which has the form \((x, y, z)\).

Fill in the blank. The determinant of the matrix obtained by deleting the \(i\)th row and \(j\)th column of a square matrix \(A\) is called the _______ of the entry \(a_{i j}\).

Fill in the blank. A message written according to a secret code is called a _______ .

Fill in the blank. A matrix in row-echelon form is in _____________ when every column that has a leading 1 has zeros in every position above and below its leading \(1 .\)

Fill in the blank(s). A_____of a system of equations is an ordered pair that satisfies each equation in the system.

Two systems of equations that have the same solution set are called _____ systems.

When working with matrices, real numbers are often referred to as ______.

For a square matrix \(B\), the minor \(M_{23}=5 .\) What is the cofactor \(C_{23}\) of matrix \(B ?\)

Do all square matrices have inverses?

Fill in the blank. The process of using row operations to write a matrix in reduced row-echelon form is called

Fill in the blank(s). The first step in solving a system of two equations in \(x\) and \(y\) by the method of ____ is to solve one of the equations for one variable in terms of the other.

Is a system of linear equations with no solution consistent or inconsistent?

A matrix consisting entirely of zeros is called a ______ matrix and is denoted by ______ .

Fill in the blank. The process used to write a system of equations in row-echelon form is called _____ elimination.

To find the determinant of a matrix using expanding by cofactors, do you need to find all the cofactors?

Consider three points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) and the determinant shown at the right. Suppose the value of the determinant is \(0 .\) What can you conclude?

Given that \(A\) and \(B\) are square matrices and \(A B=I_{n},\) does \(B A=I_{n} ?\)

Refer to the system of linear equations \(\left\\{\begin{aligned}-2 x+3 y &=5 \\\ 6 x+7 y &=4 \end{aligned}\right.\). Is the coefficient matrix for the system a square matrix?

Fill in the blank(s). A point of intersection of the graphs of the equations of a system is a _____ of the system.

Is a system of linear equations with at least one solution consistent or inconsistent?

The \(n \times n\) matrix consisting of 1 's on its main diagonal and 0 's elsewhere is called the ______ matrix of dimension \(n.\)

Fill in the blank. A system of equations is called _____ when the number of equations differs from the number of variables in the system.

Find the determinant of the matrix. $$[4]$$

Use a determinant to find the area of the figure with the given vertices. (-2,4),(2,3),(-1,5)

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rr} 1 & 3 \\ -1 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & -3 \\ 1 & 1 \end{array}\right]$$

Refer to the system of linear equations \(\left\\{\begin{aligned}-2 x+3 y &=5 \\\ 6 x+7 y &=4 \end{aligned}\right.\). Is the augmented matrix for the system of dimension \(3 \times 2 ?\)

What is the point of intersection of the graphs of the cost and revenue functions called?

Is a system of two linear equations consistent when the lines are coincident?

Match the matrix property with the correct form. \(A, B,\) and \(C\) are matrices, and \(c\) and \(d\) are scalars. (a) \((c d) A=c(d A)\) (b) \(A+B=B+A\) (c) \(1 A=A\) (d) \(c(A+B)=c A+c B\) (e) \(A+(B+C)=(A+B)+C\). (i) Commutative Property of Matrix Addition (ii) Associative Property of Matrix Addition (iii) Associative Property of Scalar Multiplication (iv) Scalar Identity

Fill in the blank. Solutions of equations in three variables can be pictured using a _____ coordinate system.

Use a determinant to find the area of the figure with the given vertices. \((-3,5),(2,6),(3,-5)\)

Find the determinant of the matrix. $$[-12]$$

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rr} -4 & 3 \\ 3 & -2 \end{array}\right], \quad B=\left[\begin{array}{ll} 2 & 3 \\ 3 & 4 \end{array}\right]$$

Refer to the system of linear equations \(\left\\{\begin{aligned}-2 x+3 y &=5 \\\ 6 x+7 y &=4 \end{aligned}\right.\). Is the augmented matrix row-equivalent to its reduced row-echelon form?

The graphs of the equations of a system do not intersect. What can you conclude about the system?

When a system of linear equations has no solution, do the lines intersect?

Match the matrix property with the correct form. \(A, B,\) and \(C\) are matrices, and \(c\) and \(d\) are scalars. (a) \(A(B+C)=A B+A C\) (b) \(c(A B)=(c A) B=A(c B)\) (c) \(A(B C)=(A B) C\) (d) \((A+B) C=A C+B C\) (i) Associative Property of Matrix Multiplication (ii) Left Distributive Property (iii) Right Distributive Property (iv) Associative Property of Scalar Multiplication

Fill in the blank. The process of writing a rational expression as the sum of two or more simpler rational expressions is called ____.

Find the determinant of the matrix. $$\left[\begin{array}{rr} 8 & 4 \\ -2 & 3 \end{array}\right]$$

Use a determinant to find the area of the figure with the given vertices. \(\left(0, \frac{1}{2}\right),\left(\frac{5}{2}, 0\right),(4,3)\)

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{ll} 5 & 4 \\ 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} -1 & 2 \\ \frac{3}{2} & -\frac{5}{2} \end{array}\right]$$

Determine the dimension of the matrix. $$\left[\begin{array}{ll} 7 & 0 \end{array}\right]$$

Determine whether each ordered pair is a solution of the system of equations. $$\left\\{\begin{array}{l} 4 x-y=1 \\ 6 x+y=-6 \end{array}\right.$$ (a) (0,-3) (b) (-1,-5) (c) \(\left(-\frac{3}{2}, 3\right)\) (d) \(\left(-\frac{1}{2},-3\right)\)