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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 25
Chapter 7, Problem 25

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣310−340−13−5∣\(\begin{vmatrix}\)3 & 1 & 0 \\-3 & 4 & 0 \\-1 & 3 & -5\(\end{vmatrix}\)

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Identify the 3x3 matrix for which you need to find the determinant using expansion by minors: \[\begin{vmatrix} 3 & 1 & 0 \\ -3 & 4 & 0 \\ -1 & 3 & -5 \end{vmatrix}\]
Choose a row or column to expand along. It is often easiest to choose a row or column with zeros to simplify calculations. Here, the third column has two zeros, so expand along the third column.
Write the determinant as a sum of the elements in the chosen column multiplied by their corresponding cofactors. For the third column, the determinant is: \[0 \cdot C_{13} + 0 \cdot C_{23} + (-5) \cdot C_{33}\] where \(C_{ij}\) is the cofactor of the element in row \(i\), column \(j\).
Calculate the cofactor \(C_{33}\). The cofactor is given by: \[C_{33} = (-1)^{3+3} \times M_{33}\] where \(M_{33}\) is the minor obtained by deleting the third row and third column from the matrix. So, find the determinant of the 2x2 matrix: \[\begin{vmatrix} 3 & 1 \\ -3 & 4 \end{vmatrix}\]
Compute the 2x2 determinant: \[M_{33} = (3)(4) - (1)(-3)\] Then multiply by \((-1)^{6} = 1\) to get \(C_{33}\). Finally, multiply \(C_{33}\) by the element \(-5\) from the original matrix to get the determinant of the 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 a square matrix and provides important properties such as invertibility. For a 3x3 matrix, the determinant helps determine if the matrix is singular or nonsingular and is essential in solving systems of linear equations.
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Expansion by Minors

Expansion by minors is a method to calculate the determinant of a matrix by expanding along a row or column. It involves computing minors, which are determinants of smaller matrices formed by deleting one row and one column, and combining them with cofactors that include sign adjustments.
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Cofactor and Sign Pattern

A cofactor is the minor of an element multiplied by (-1) raised to the sum of the element's row and column indices. The sign pattern alternates in a checkerboard fashion starting with a positive sign at the top-left element, which is crucial for correctly calculating the determinant using expansion by minors.
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Related Practice
Textbook Question

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣111222−34−5∣\(\begin{vmatrix}\)1 & 1 & 1 \\2 & 2 & 2 \\-3 & 4 & -5\(\end{vmatrix}\)

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Textbook Question

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

{x+y+z=4x−y−z=0x−y+z=2\(\begin{cases}\)x + y + z = 4 \(\x\) - y - z = 0 \(\x\) - y + z = 2\(\end{cases}\)

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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+2x+3y−z=72x−3y+z=4w−4x+y=3\(\begin{cases}\)w + 2x + 3y - z = 7 \\2x - 3y + z = 4 \(\w\) - 4x + y = 3\(\end{cases}\)

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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. 4A + 3B = - 2X

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Textbook Question

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

{2x−y−z=4x+y−5z=−4x−2y=4\(\begin{cases}\)2x - y - z = 4 \(\x\) + y - 5z = -4 \(\x\) - 2y = 4\(\end{cases}\)

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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. B - X = 4A

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