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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 41
7장, 문제 41

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.
A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)
A - C

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Step 1: Identify the dimensions of matrices A and C. Matrix A is a 3x2 matrix (3 rows, 2 columns), and matrix C is a 2x2 matrix (2 rows, 2 columns).
Step 2: To perform the subtraction A - C, the matrices must have the same dimensions. Since A is 3x2 and C is 2x2, their dimensions do not match.
Step 3: Because the dimensions of A and C are different, the operation A - C is not defined. Matrix addition or subtraction requires both matrices to have the same number of rows and columns.
Step 4: Conclude that the operation A - C cannot be performed due to incompatible dimensions.
Step 5: If you want to perform matrix operations, consider checking other pairs of matrices with matching dimensions or perform multiplication if dimensions allow.

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주요 개념

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Matrix Addition and Subtraction

Matrix addition and subtraction require matrices to have the same dimensions. Each element in one matrix is added to or subtracted from the corresponding element in the other matrix. If the matrices differ in size, the operation is undefined.
추천 영상:
03:18
Adding and Subtracting Complex Numbers

Matrix Dimensions and Compatibility

Understanding the dimensions (rows × columns) of matrices is crucial for determining if operations like addition, subtraction, or multiplication are defined. For example, subtraction requires identical dimensions, while multiplication requires the number of columns in the first matrix to equal the number of rows in the second.
추천 영상:
가이드 코스
4:35
Introduction to Matrices

Element-wise Operations

In operations like A - C, each element of matrix C is subtracted from the corresponding element of matrix A. This requires careful alignment of elements based on their position, reinforcing the need for matrices to be conformable in size.
추천 영상:
가이드 코스
8:38
Performing Row Operations on Matrices
관련 실천
교과서 질문

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

A(BC)

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교과서 질문

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

{2lnw+lnx+3lny−2lnz=−64lnw+3lnx+lny−lnz=−2lnw+lnx+lny+lnz=−5lnw+lnx−lny−lnz=5\(\begin{cases}\) 2 \(\ln\) w + \(\ln\) x + 3 \(\ln\) y - 2 \(\ln\) z = -6 \\ 4 \(\ln\) w + 3 \(\ln\) x + \(\ln\) y - \(\ln\) z = -2 \\ \(\ln\) w + \(\ln\) x + \(\ln\) y + \(\ln\) z = -5 \\ \(\ln\) w + \(\ln\) x - \(\ln\) y - \(\ln\) z = 5 \(\end{cases}\)

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교과서 질문

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

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교과서 질문

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

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교과서 질문

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=4x−2y+z=7x+3y+2z=4\(\begin{cases}\) x + y + z = 4 \\ x - 2y + z = 7 \\ x + 3y + 2z = 4 \(\end{cases}\)

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교과서 질문

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{w−x+2y=−3x−y+z=4−w+x−y+2z=2−x+y−2z=−4The inverse of [1−12001−11−11−120−11−2] is [00−1−1141312120−10−1]\(\begin{cases}\) w - x + 2y \(\quad\]\quad\) = -3 \\ \(\quad\[\quad\) x - y + z = 4 \\ -w + x - y + 2z = 2 \\ \(\quad\]\quad\) -x + y - 2z = -4 \(\end{cases}\) \\ \(\text{The inverse of }\) \(\begin{bmatrix}\) 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \(\end{bmatrix}\) \(\text{ is }\) \(\begin{bmatrix}\) 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \(\end{bmatrix}\)

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