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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 23
Capitolo 7, Problema 23

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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Start with the given matrix equation: \(B - X = 4A\).
To isolate \(X\), subtract \(B\) from both sides and multiply by \(-1\): \(X = B - 4A\).
Calculate \$4A$ by multiplying each element of matrix $A$ by 4: \(4A = 4 \times \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix}\).
Perform the matrix subtraction \(B - 4A\) by subtracting corresponding elements of \$4A$ from $B$: \(X = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix} - 4A\).
Write the resulting matrix \(X\) with the calculated values from the subtraction.

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Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same dimensions. Each element in the resulting matrix is found by adding or subtracting the elements in the same position from the original matrices. This operation is fundamental for solving equations involving matrices.
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Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every element of a matrix by a constant (scalar). This operation scales the matrix and is essential when manipulating matrix equations, such as multiplying matrix A by 4 in the given problem.
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Solving Matrix Equations

Solving matrix equations requires isolating the unknown matrix by performing inverse operations, similar to algebraic equations. For the equation B - X = 4A, you can isolate X by subtracting 4A from B or rearranging terms, ensuring all operations respect matrix dimensions.
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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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Domanda del libro di testo

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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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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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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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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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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