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

Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 41

Chapter 7, Problem 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.
$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$
A - C

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

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Textbook Question

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.

$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$

A(BC)

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Textbook Question

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.)

$\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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Textbook Question

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

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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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Textbook Question

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

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Textbook Question

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.

$\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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