Chapter 8

The method of using determinants to solve a system of linear equations is called ________ ________.

Both det(\(A\)) and |\(A\)| represent the ________ of the matrix \(A\).

In a ________ matrix, the number of rows equals the number of columns.

Two matrices are ________ if all of their corresponding entries are equal.

A rectangular array of real numbers that can be used to solve a system of linear equations is called a ________.

Three points are ________ if the points lie on the same line.

The ________ \(M_{ij}\) of the entry \(a_{ij}\) is the determinant of the matrix obtained by deleting the \(i\)th row and \(j\)th column of the square matrix \(A\).

If there exists an \(n \times n\) matrix \(A^{-1}\) such that \(AA^{-1} = I_n = A^{-1}A\), then \(A^{-1}\) is called the ________ of \(A\).

When performing matrix operations, real numbers are often referred to as ________.

A matrix is ________ if the number of rows equals the number of columns.

The area \(A\) of a triangle with vertices \((x_1, y_1),\) \((x_2, y_2),\) and \((x_3, y_3)\) is given by ________.

The ________ \(C_{ij}\) of the entry \(a_{ij}\) of the square matrix \(A\) is given by \((-1)^{i+j}M_{ij}\).

If a matrix \(A\) has an inverse, it is called invertible or ________; if it does not have an inverse, it is called ________.

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

For a square matrix, the entries \(a_{11}\), \(a_{22}\), \(a_{33}\), \(\ldots\), \(a_{nn}\) are the ________ ________ entries.

A message written according to a secret code is called a ________.

If \(A\) is an invertible matrix, the system of linear equations represented by \(AX=B\) has a unique solution given by \(X =\) ________.

The \(n\) x \(n\) matrix consisting of 1's on its main diagonal and 0's elsewhere is called the ________ matrix of order \(n\) x \(n\).

A matrix with only one row is called a ________ matrix, and a matrix with only one column is called a ________ matrix.

In Exercises 5-20, find the determinant of the matrix. \([4]\)

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 2 && 1 \\ 5 && 3 \end{array} \right]\), \(B = \left[ \begin{array}{r} 3 & -1 \\ -5 & 2 \end{array} \right]\).

In Exercises 5 and 6, match the matrix property with the correct form. \(A\), \(B\), and \(C\) are matrices of order and and are scalars. (a) 1\(A = A\) (b) \(A + (B+C) = (A+B) + C\) (c) \((c+d)A = cA + dA\) (d) \((cd)A = c(dA)\) (e) \(A + B = B +A\) (i) Distributive Property (ii) Commutative Property of Matrix Addition (iii) Scalar Identity Property (iv) Associative Property of Matrix Addition (v) Associative Property of Scalar Multiplication

The matrix derived from a system of linear equations is called the ________ matrix of the system.

If a message is encoded using an invertible matrix \(A\), then the message can be decoded by multiplying the coded row matrices by ________ (on the right).

In Exercises 5-20, find the determinant of the matrix. \([-10]\)

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 1 & -1 \\ -1 & 2 \end{array} \right]\), \(B = \left[ \begin{array}{r} 2 && 1 \\ 1 && 1 \end{array} \right]\).

In Exercises 5 and 6, match the matrix property with the correct form. \(A\), \(B\), and \(C\) are matrices of order and and are scalars. (a) \(A + O = A\) (b) \(c(AB) = A(cB)\) (c) \(A(B + C) = AB + AC\) (d) \(A(BC) = (AB)C\) (i) Distributive Property (ii) Additive Identity of Matrix Addition (iii) Associative Property of Matrix Multiplication (iv) Associative Property of Scalar Multiplication

The matrix derived from the coefficients of a system of linear equations is called the ________ matrix of the system.

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} -7x + 11y = -1 \\ 3x - 9y = 9 \end{cases}\)

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 8 & 4 \\ 2 & 3 \end{array} \right]\)

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 1 && 2 \\ 3 && 4 \end{array} \right]\), \(B = \left[ \begin{array}{r} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{array} \right]\).

In Exercises 7-10, find \(x\) and \(y\). \(\left[ \begin{array}{r} x & -2 \\ 7 & y \end{array} \right] = \left[ \begin{array}{r} -4 & -2 \\ 7 & 22 \end{array} \right]\)

Two matrices are called ________ if one of the matrices can be obtained from the other by a sequence of elementary row operations.

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} 4x - 3y = -10 \\ 6x + 9y = 12 \end{cases}\)

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} -9 & 0 \\ 6 & 2 \end{array} \right]\)

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 1 & -1 \\ 2 & 3 \end{array} \right]\), \(B = \left[ \begin{array}{r} \frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5} \end{array} \right]\).

In Exercises 7-10, find \(x\) and \(y\). \(\left[ \begin{array}{r} -5 & x \\ y & 8 \end{array} \right] = \left[ \begin{array}{r} -5 & 13 \\ 12 & 8 \end{array} \right]\)

A matrix in row-echelon form is in ________ ________ ________ if every column that has a leading 1 has zeros in every position above and below its leading 1.

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 6 & 2 \\ -5 & 3 \end{array} \right]\)

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 2 & -17 & 11 \\ -1 & 11 & -7 \\ 0 & 3 & -2 \end{array} \right]\), \(B = \left[ \begin{array}{r} 1 & 1 & 2 \\ 2 & 4 & -3 \\\ 3 & 6 & -5 \end{array} \right]\).

In Exercises 7-10, find \(x\) and \(y\). \(\left[ \begin{array}{r} 16 & 4 & 5 & 4 \\ -3 & 13 & 15 & 6 \\ 0 & 2 & 4 & 0 \end{array} \right] = \left[ \begin{array}{r} 16 & 4 & 2x+1 & 4 \\ -3 & 13 & 15 & 3x \\ 0 & 2 & 3y-5 & 0 \end{array} \right]\)

In Exercises 9-14, determine the order of the matrix. \( \left[\begin{array}{rr} 7 & & 0 \end{array}\right] \)

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} 6x - 5y = 17 \\ -13x + 3y = -76 \end{cases}\)

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 3 & -3 \\ 4 & -8 \end{array} \right]\)

In Exercises \(5-12,\) show that \(B\) is the inverse of \(A .\) $$A=\left[\begin{array}{rrr}{-4} & {1} & {5} \\ {-1} & {2} & {4} \\ {0} & {-1} & {-1}\end{array}\right], B=\left[\begin{array}{rrr}{-\frac{1}{2}} & {1} & {\frac{3}{2}} \\ {\frac{1}{4}} & {-1} & {-\frac{11}{4}} \\ {-\frac{1}{4}} & {1} & {\frac{7}{4}}\end{array}\right]$$

In Exercises 7-10, find \(x\) and \(y\). \(\left[ \begin{array}{r} x+2 & 8 & -3 \\ 1 & 2y & 2x \\ 7 & -2 & y+2 \end{array} \right] = \left[ \begin{array}{r} 2x+6 & 8 & -3 \\ 1 & 18 & -8 \\\ 7 & -2 & 11 \end{array} \right]\)

In Exercises 9-14, determine the order of the matrix. \( \left[\begin{array}{rrrr} 5 & & -3 & & 8 & & 7 \end{array}\right] \)

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} -0.4x + 0.8y = 1.6 \\ 0.2x + 0.3y = 2.2 \end{cases}\)

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} -7 && 0 \\ 3 && 0 \end{array} \right]\)

In Exercises \(5-12,\) show that \(B\) is the inverse of \(A .\) $$A=\left[\begin{array}{rrr}{-2} & {2} & {3} \\ {1} & {-1} & {0} \\ {0} & {1} & {4}\end{array}\right], B=\frac{1}{3}\left[\begin{array}{rrr}{-4} & {-5} & {3} \\ {-4} & {-8} & {3} \\ {1} & {2} & {0}\end{array}\right]$$