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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 6 - Matrices and Determinants문제 5
7장, 문제 5

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x4]=[6y]\(\begin{bmatrix}\) x \\ 4 \(\end{bmatrix}\) = \(\begin{bmatrix}\) 6 \\ y \(\end{bmatrix}\)

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Understand that two matrices are equal if and only if their corresponding entries are equal.
Set the corresponding elements of the matrices equal to each other: \(x = 6\) and \(4 = y\).
From the first equation, identify the value of \(x\) as 6.
From the second equation, identify the value of \(y\) as 4.
Conclude that the values of the variables that make the matrices equal are \(x = 6\) and \(y = 4\).

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주요 개념

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Matrix Equality

Two matrices are equal if and only if they have the same dimensions and their corresponding entries are equal. This means each element in one matrix must match the element in the same position in the other matrix.
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가이드 코스
4:35
Introduction to Matrices

Solving for Variables in Matrices

When matrices contain variables, you can find their values by setting corresponding elements equal to each other. This creates simple equations that can be solved individually to determine the unknown variables.
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가이드 코스
4:35
Introduction to Matrices

Basic Algebraic Manipulation

Solving for variables often requires basic algebraic skills such as isolating the variable, performing inverse operations, and simplifying expressions. These skills are essential to find the values that satisfy the matrix equality.
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가이드 코스
05:09
Introduction to Algebraic Expressions
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Write the augmented matrix for each system of linear equations.

{5x−2y−3z=0x+y=52x−3z=4\(\begin{cases}\) 5x - 2y - 3z = 0 \\ x + y = 5 \\ 2x - 3z = 4 \(\end{cases}\)

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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. {3x+4y+2z=34x−2y−8z=−4x+y−z=3\(\begin{cases}\) 3x + 4y + 2z = 3 \\ 4x - 2y - 8z = -4 \\ x + y - z = 3 \(\end{cases}\)

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∣−7142−4∣\(\begin{vmatrix}\) -7 & 14 \\ 2 & -4 \(\end{vmatrix}\)

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

{3x1+5x2−8x3+5x4=−8x1+2x2−3x3+x4=−72x1+3x2−7x3+3x4=−114x1+8x2−10x3+7x4=−10\(\begin{cases}\) 3x_1 + 5x_2 - 8x_3 + 5x_4 = -8 \\ x_1 + 2x_2 - 3x_3 + x_4 = -7 \\ 2x_1 + 3x_2 - 7x_3 + 3x_4 = -11 \\ 4x_1 + 8x_2 - 10x_3 + 7x_4 = -10 \(\end{cases}\)

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Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A=[−2132−12],B=[1234]A = \(\begin{bmatrix}\) -2 & 1 \\ \(\frac{3}{2}\) & -\(\frac{1}{2}\) \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 \\ 3 & 4 \(\end{bmatrix}\)

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∣−5−1−2−7∣\(\begin{vmatrix}\) -5 & -1 \\ -2 & -7 \(\end{vmatrix}\)

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