Skip to main content
Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 13th EditionCh. 5 - Systems and Matrices문제 83
6장, 문제 83

Find the inverse, if it exists, for each matrix.
[2153]\(\left\)[ \(\begin{matrix}\) 2 & 1 \\ 5 & 3 \(\end{matrix}\) \(\right\)]

검증된 단계별 안내
1
Identify the given 2x2 matrix as \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
Calculate the determinant of matrix \( A \) using the formula \( \det(A) = ad - bc \).
Check if the determinant \( \det(A) \) is nonzero. If it is zero, the inverse does not exist.
If the determinant is nonzero, find the inverse matrix using the formula \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
Write the inverse matrix explicitly by substituting the values of \( a, b, c, d \) and the determinant into the formula.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Matrix Inverse

The inverse of a matrix A is another matrix, denoted A⁻¹, such that when multiplied together, they yield the identity matrix. Only square matrices with nonzero determinants have inverses. Finding the inverse is essential for solving matrix equations and understanding linear transformations.
추천 영상:
4:30
Graphing Logarithmic Functions

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. It indicates whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides information about the matrix's scaling and orientation effects.
추천 영상:
가이드 코스
4:36
Determinants of 2×2 Matrices

Formula for the Inverse of a 2x2 Matrix

For a 2x2 matrix [[a, b], [c, d]] with nonzero determinant, the inverse is (1/det) times the matrix [[d, -b], [-c, a]]. This formula swaps the diagonal elements, changes the signs of the off-diagonal elements, and scales by the reciprocal of the determinant.
추천 영상:
06:36
Solving Quadratic Equations Using The Quadratic Formula
관련 실천
교과서 질문

For each pair of matrices A and B, find (a) AB and (b) BA.

A=[−101011−1−10],B=[001010100]A = \(\left\)[ \(\begin{matrix}\) -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \(\end{matrix}\) \(\right\)]

112
views
교과서 질문

Find AB and BA for the following matrices.

A=[abcd]andB=[1001]A = \(\left\)[ \(\begin{matrix}\) a & b \\ c & d \(\end{matrix}\) \(\right\)] \(\quad\) \(\text{and}\) \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 \\ 0 & 1 \(\end{matrix}\) \(\right\)]

Matrix B acts as the multiplicative element for 2 ×\(\times\) 2 square matrices.

131
views
교과서 질문

Perform each operation, if possible.

[−2550143−4−1][10−1−10011−1]\(\left\)[ \(\begin{matrix}\) -2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & -4 & -1 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 1 & 0 & -1 \\ -1 & 0 & 0 \\ 1 & 1 & -1 \(\end{matrix}\) \(\right\)]

83
views
교과서 질문

For each pair of matrices A and B, find (a) AB and (b) BA. A=[01−1010001],B=[100010001]A = \(\left\)[ \(\begin{matrix}\) 0 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)]

62
views
교과서 질문

Solve each problem. See Examples 5 and 9. The sum of two numbers is 47, and the difference between the numbers is 1. Find the numbers.

698
views
교과서 질문

Find the inverse, if it exists, for each matrix.

[2−101011−20]\(\left\)[ \(\begin{matrix}\) 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & -2 & 0 \(\end{matrix}\) \(\right\)]

137
views