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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 21
Capitolo 7, Problema 21

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

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1
Start with the given matrix equation: \(3X + 2A = B\).
Isolate the term with \(X\) by subtracting \$2A$ from both sides: \(3X = B - 2A\).
To solve for \(X\), divide both sides of the equation by 3, which is equivalent to multiplying by \(\frac{1}{3}\): \(X = \frac{1}{3}(B - 2A)\).
Calculate the matrix \$2A$ by multiplying each element of matrix $A$ by 2.
Subtract the matrix \$2A$ from matrix $B$ element-wise, then multiply the resulting matrix by \(\frac{1}{3}\) to find matrix $X$.

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Matrix Addition and Scalar Multiplication

Matrix addition involves adding corresponding elements of two matrices of the same size. Scalar multiplication means multiplying every element of a matrix by a constant. These operations are essential to manipulate the equation 3X + 2A = B by distributing scalars and combining matrices.
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Solving Matrix Equations

To solve matrix equations like 3X + 2A = B, isolate the matrix variable X by performing inverse operations. This typically involves subtracting 2A from both sides and then multiplying by the inverse of the scalar coefficient (here, dividing by 3) to find X.
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Matrix Dimensions and Compatibility

Matrix operations require matrices to have compatible dimensions. Both A and B are 3x2 matrices, so X must also be 3x2 for the equation to be valid. Understanding dimensions ensures correct addition and scalar multiplication without errors.
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Introduction to Matrices
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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+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 \(\end{cases}\)

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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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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. 2X + A = B

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