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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 23
Chapter 7, Problem 23

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣30021−525−1∣\(\begin{vmatrix}\)3 & 0 & 0 \\2 & 1 & -5 \\2 & 5 & -1\(\end{vmatrix}\)

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Identify the matrix for which you need to find the determinant: \[\begin{bmatrix} 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \end{bmatrix}\]
Choose a row or column to expand by minors. Since the first row has two zeros, expanding along the first row is efficient.
Write the determinant expansion along the first row: \[\text{det} = 3 \cdot C_{11} + 0 \cdot C_{12} + 0 \cdot C_{13}\] where \(C_{ij}\) is the cofactor of the element in row \(i\), column \(j\).
Calculate the cofactor \(C_{11}\) by finding the determinant of the 2x2 submatrix obtained by removing the first row and first column: \[\begin{bmatrix} 1 & -5 \\ 5 & -1 \end{bmatrix}\] The determinant of this submatrix is calculated as \((1)(-1) - (-5)(5)\).
Multiply the element \(3\) by the cofactor \(C_{11}\) to get the determinant of the original 3x3 matrix.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Determinant of a Matrix

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties of the matrix, such as invertibility, and is used in solving systems of linear equations. For a 3x3 matrix, the determinant can be found using expansion by minors or other methods.
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Determinants of 2×2 Matrices

Expansion by Minors

Expansion by minors is a method to calculate the determinant of a matrix by breaking it down into smaller determinants of submatrices. This involves selecting a row or column, multiplying each element by the determinant of its minor matrix, and applying alternating signs. It simplifies the calculation of larger determinants.
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Minor and Cofactor

A minor is the determinant of the smaller matrix formed by deleting one row and one column from the original matrix. The cofactor is the minor multiplied by (-1)^(row+column), which accounts for sign changes in expansion by minors. Understanding minors and cofactors is essential for correctly applying expansion by minors.
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Related Practice
Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w+x−y+z=−22w−x+2y−z=7−w+2x+y+2z=−1\(\begin{cases}\)w + x - y + z = -2 \\2w - x + 2y - z = 7 \\-w + 2x + y + 2z = -1\(\end{cases}\)

Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{x+y−z=−22x−y+z=5−x+2y+2z=1\(\begin{cases}\)x + y - z = -2 \\2x - y + z = 5 \\-x + 2y + 2z = 1\(\end{cases}\)

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Textbook Question

Perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. BD

A=[2−1253−1]B=[0−2321−5]C=[123−112−121]D=[−2313−24]A=\(\begin{bmatrix}\)2 & -1 & 2\\ 5 & 3 & -1\(\end{bmatrix}\[\quad\) B=\(\begin{bmatrix}\)0 & -2\\ 3 & 2\\ 1 & -5\(\end{bmatrix}\)C=\(\begin{bmatrix}\)1 & 2 & 3\\ -1 & 1 & 2\\ -1 & 2 & 1\(\end{bmatrix}\]\quad\) D=\(\begin{bmatrix}\)-2 & 3 & 1\\ 3 & -2 & 4\(\end{bmatrix}\)

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Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{x+3y=0x+y+z=13x−y−z=11\(\begin{cases}\)x + 3y = 0 \(\x\) + y + z = 1 \\3x - y - z = 11\(\end{cases}\)

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Textbook Question

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. 3X + 2A = B

Textbook Question

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. B - X = 4A