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Chapter 16: Problem 16

Find, if possible, A B and B A. If it is not possible. explain why. $$A=\left[\begin{array}{rrr} -3 & 2 & 0 \\ 1 & -4 & 5 \end{array}\right] \quad B=\left[\begin{array}{rr} -2 & 0 \\ 4 & -6 \\ 5 & -1 \end{array}\right]$$

Short Answer

Expert verified
A B is defined and equals \( \left[\begin{array}{rr} 14 & -12 \\ 7 & 19 \end{array}\right] \), B A is defined and equals \( \left[\begin{array}{rrr} 6 & -4 & 0 \\ -18 & 32 & -30 \\ -16 & 14 & -5 \end{array}\right] \).

Step by step solution

01

Understand Matrix Multiplication

Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second matrix. For two matrices, say matrix \( C \) of size \( m \times n \) and matrix \( D \) of size \( p \times q \), the product matrix \( CD \) is defined if and only if \( n = p \), resulting in a product matrix of size \( m \times q \).
02

Check Compatibility for A B

Matrix \( A \) is of size \( 2 \times 3 \) and matrix \( B \) is of size \( 3 \times 2 \). The number of columns in \( A \) is equal to the number of rows in \( B \) (3 = 3), so the multiplication \( AB \) is possible. The resulting matrix will be of size \( 2 \times 2 \).
03

Calculate A B

To find \( AB \), perform the following calculations:1. First row, first column: \((-3)(-2) + (2)(4) + (0)(5) = 6 + 8 + 0 = 14\)2. First row, second column: \((-3)(0) + (2)(-6) + (0)(-1) = 0 - 12 + 0 = -12\)3. Second row, first column: \((1)(-2) + (-4)(4) + (5)(5) = -2 - 16 + 25 = 7\)4. Second row, second column: \((1)(0) + (-4)(-6) + (5)(-1) = 0 + 24 - 5 = 19\)Thus, \( AB = \left[\begin{array}{rr} 14 & -12 \ 7 & 19 \end{array}\right] \).
04

Check Compatibility for B A

Matrix \( B \) is of size \( 3 \times 2 \) and matrix \( A \) is of size \( 2 \times 3 \). Here, the number of columns in \( B \) equals the number of rows in \( A \) (2 = 2), so the multiplication \( BA \) is possible. The resulting matrix will be of size \( 3 \times 3 \).
05

Calculate B A

To compute \( BA \), perform the following calculations:1. First row: - First column: \((-2)(-3) + (0)(1) = 6 + 0 = 6\) - Second column: \((-2)(2) + (0)(-4) = -4 + 0 = -4\) - Third column: \((-2)(0) + (0)(5) = 0 + 0 = 0\)2. Second row: - First column: \((4)(-3) + (-6)(1) = -12 - 6 = -18\) - Second column: \((4)(2) + (-6)(-4) = 8 + 24 = 32\) - Third column: \((4)(0) + (-6)(5) = 0 - 30 = -30\)3. Third row: - First column: \((5)(-3) + (-1)(1) = -15 - 1 = -16\) - Second column: \((5)(2) + (-1)(-4) = 10 + 4 = 14\) - Third column: \((5)(0) + (-1)(5) = 0 - 5 = -5\)Thus, \( BA = \left[\begin{array}{rrr} 6 & -4 & 0 \ -18 & 32 & -30 \ -16 & 14 & -5 \end{array}\right] \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Dimensions
Matrix dimensions are the foundation for understanding all matrix operations, such as multiplication. When we talk about matrix dimensions, we refer to the number of rows and columns in a matrix. For example, a matrix with 2 rows and 3 columns is said to be of size \(2 \times 3\). In matrix notation, rows are counted vertically, and columns are counted horizontally.
Understanding matrix dimensions is crucial because they determine how matrices can interact with each other through operations like addition, subtraction, and especially multiplication.
Compatibility of Matrices
In matrix multiplication, compatibility refers to whether the operation can be performed between two matrices. For this, it is essential that the number of columns in the first matrix equals the number of rows in the second matrix. If matrix \(A\) is of size \(m \times n\) and matrix \(B\) is of size \(p \times q\), then the operation \(AB\) will only be possible if \(n = p\). This compatibility condition ensures that each element in the resulting matrix is a valid computation.
Think of compatibility as a gatekeeper: if the dimensions do not meet the specific requirement, the multiplication cannot be carried out.
Resulting Matrix Size
When matrices are compatible for multiplication, the size of the resulting matrix is determined by the number of rows from the first matrix and the number of columns from the second matrix. For matrices \(C\) and \(D\), if the multiplication \(CD\) is viable, the resultant matrix will have dimensions \(m \times q\).
For example, multiplying a \(2 \times 3\) matrix with a \(3 \times 2\) matrix results in a \(2 \times 2\) matrix. These dimensions inform us about the structural layout of the resultant matrix, which helps in understanding the full scope of the multiplication process.
Dot Product in Matrices
The dot product in matrices is a crucial operation that helps generate the values for the resulting matrix in multiplication. This process involves taking each row in the first matrix and multiplying the corresponding elements with each column in the second matrix, then summing the results.
For instance, to find an element located at the first row and first column of the resulting matrix \(AB\), you compute the dot product of the entire first row of matrix \(A\) with the first column of matrix \(B\). Repeat this process for each row and column to fill out the entire resulting matrix. This operation underpins the whole matrix multiplication concept and requires careful matching of specific elements.

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Most popular questions from this chapter

Set up systems of equations and solve by Gaussian elimination. The voltage across an electric resistor equals the current (in A) times the resistance (in \(\Omega\) ). If a current of 3.00 A passes through each of two resistors, the sum of the voltages is 10.5 V. If \(2.00 \mathrm{A}\) passes through the first resistor and 4.00 A passes through the second resistor, the sum of the voltages is \(13.0 \mathrm{V}\). Find the resistances.

Find the inverse of each of the given matrices by using a calculator. $$\left[\begin{array}{rrr} 1 & 3 & 2 \\ -2 & -5 & -1 \\ 2 & 4 & 0 \end{array}\right]$$

Solve the given problems using Gaussian elimination. Solve the system \(x+2 y=6,2 x+a y=4\) and show that the solution depends on the value of \(a\). What value of \(a\) does the solution show may not be used?

Perform the indicated matrix operations. The inventory of a drug supply company shows that the following numbers of cases of bottles of vitamins \(\mathrm{C}\) and \(\mathrm{B}_{3}\) (niacin) are in stock: Vitamin \(\mathrm{C}-25\) cases of \(100-\mathrm{mg}\) bottles, 10 cases of \(250-\mathrm{mg}\) bottles, and 32 cases of 500 -mg bottles; vitamin \(\mathrm{B}_{3}-30\) cases of \(100-\mathrm{mg}\) bottles, 18 cases of \(250-\mathrm{mg}\) bottles, and 40 cases of \(500-\mathrm{mg}\) bottles. This is represented by matrix \(A\) below. After two shipments are sent out, each of which can be represented by matrix \(B\) below, find the matrix that represents the remaining inventory. $$A=\left[\begin{array}{lll} 25 & 10 & 32 \\ 30 & 18 & 40 \end{array}\right] \quad B=\left[\begin{array}{lll} 10 & 5 & 6 \\ 12 & 4 & 8 \end{array}\right]$$

Determine by matrix multiplication whether or not A is the proper matrix of solution values. $$\begin{array}{l} 2 x-y+z=7 \\ x-3 y+2 z=7 \\ 3 x+y=7 \end{array} \quad A=\left[\begin{array}{r} 3 \\ -2 \\ -1 \end{array}\right]$$
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