Question

For \((a)\) through \((d)\) find \(C=A B\) \((a) A=\left[\begin{array}{rr}12 & 14 \\ 20 & 5\end{array}\right] \quad B=\left[\begin{array}{ll}3 & 9 \\ 0 & 2\end{array}\right]\) (b) \(A=\left[\begin{array}{ll}4 & 7 \\ 9 & 1\end{array}\right] \quad B=\left[\begin{array}{lll}3 & 8 & 5 \\ 2 & 6 & 7\end{array}\right]\) (c) \(A=\left[\begin{array}{rr}7 & 11 \\ 2 & 9 \\ 10 & 6\end{array}\right] \quad B=\left[\begin{array}{rrr}12 & 4 & 5 \\ 3 & 6 & 1\end{array}\right]\) \((d) A=\left[\begin{array}{lll}6 & 2 & 5 \\ 7 & 9 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}10 & 1 \\ 11 & 3 \\ 2 & 9\end{array}\right]\) (e) Find (i) \(C=A B,\) and (ii) \(D=B A\), if \(A=\left[\begin{array}{r}-2 \\ 4 \\ 7\end{array}\right] \quad B=\left[\begin{array}{lll}3 & 6 & -2\end{array}\right]\)

Short answer

Matrix products: (a) \([\begin{matrix} 36 & 142 \\ 60 & 190 \end{matrix}]\), (b) \([\begin{matrix} 26 & 70 & 69 \\ 29 & 78 & 52 \end{matrix}]\), (c) \([\begin{matrix} 117 & 86 & 46 \\ 51 & 62 & 19 \\ 132 & 76 & 56 \end{matrix}]\), (d) \([\begin{matrix} 102 & 63 \\ 179 & 70 \end{matrix}]\), (e)(i) \([\begin{matrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{matrix}]\), (e)(ii) 0.

Step-by-step solution

Verify Matrix Dimensions

First, verify if the multiplication of matrices is possible by checking dimensions. For multiplying matrices \(A\) and \(B\), the number of columns in \(A\) must match the number of rows in \(B\).

Calculate Multiplication for (a)

Given matrices are \(A=\left[\begin{array}{rr}12 & 14 \ 20 & 5\end{array}\right]\) and \(B=\left[\begin{array}{ll}3 & 9 \ 0 & 2\end{array}\right]\). Multiply the first row of \(A\) with columns of \(B\), similarly for the second row:\[C=\left[\begin{array}{cc}(12\times3 + 14\times0) & (12\times9 + 14\times2) \ (20\times3 + 5\times0) & (20\times9 + 5\times2)\end{array}\right]=\left[\begin{array}{ll}36 & 142 \ 60 & 190\end{array}\right]\].

Calculate Multiplication for (b)

Given \(A=\left[\begin{array}{ll}4 & 7 \ 9 & 1\end{array}\right]\) and \(B=\left[\begin{array}{lll}3 & 8 & 5 \ 2 & 6 & 7\end{array}\right]\). Multiply row 1 of \(A\) with columns of \(B\), and do the same for row 2:\[C=\left[\begin{array}{ccc}(4\times3 + 7\times2) & (4\times8 + 7\times6) & (4\times5 + 7\times7) \ (9\times3 + 1\times2) & (9\times8 + 1\times6) & (9\times5 + 1\times7)\end{array}\right]=\left[\begin{array}{lll}26 & 70 & 69 \ 29 & 78 & 52\end{array}\right]\].

Calculate Multiplication for (c)

For \(A=\left[\begin{array}{rr}7 & 11 \ 2 & 9 \ 10 & 6\end{array}\right]\) and \(B=\left[\begin{array}{rrr}12 & 4 & 5 \ 3 & 6 & 1\end{array}\right]\), multiply each row of \(A\) with each column of \(B\): \[C=\left[\begin{array}{ccc}(7\times12 + 11\times3) & (7\times4 + 11\times6) & (7\times5 + 11\times1) \ (2\times12 + 9\times3) & (2\times4 + 9\times6) & (2\times5 + 9\times1) \ (10\times12 + 6\times3) & (10\times4 + 6\times6) & (10\times5 + 6\times1) \end{array}\right]\].

Simplify Results of (c)

Continuing from Step 4:\[C=\left[\begin{array}{ccc}117 & 86 & 46 \ 51 & 62 & 19 \ 132 & 76 & 56 \end{array}\right]\].

Calculate Multiplication for (d)

Given \(A=\left[\begin{array}{lll}6 & 2 & 5 \ 7 & 9 & 4\end{array}\right]\) and \(B=\left[\begin{array}{rr}10 & 1 \ 11 & 3 \ 2 & 9\end{array}\right]\), calculate the multiplication:\[C=\left[\begin{array}{cc}(6\times10 + 2\times11 + 5\times2) & (6\times1 + 2\times3 + 5\times9) \ (7\times10 + 9\times11 + 4\times2) & (7\times1 + 9\times3 + 4\times9)\end{array}\right]\].

Simplify Results of (d)

From Step 6, simplify to get:\[C=\left[\begin{array}{cc}102 & 63 \ 179 & 70\end{array}\right]\].

Determine Matrices Order for (e)

For (e), we're given vectors: \(A=\left[\begin{array}{r}-2 \ 4 \ 7\end{array}\right]\) and \(B=\left[\begin{array}{lll}3 & 6 & -2\end{array}\right]\). Matrix \(A\) is a column vector (3x1) and \(B\) is a row vector (1x3).

Calculate (i) Product C for (e)

The product \(C=AB\) is a 3x3 matrix:\[C=A B=\left[\begin{array}{ccc}(-2\times3) & (-2\times6) & (-2\times-2) \ (4\times3) & (4\times6) & (4\times-2) \ (7\times3) & (7\times6) & (7\times-2)\end{array}\right]=\left[\begin{array}{ccc}-6 & -12 & 4 \ 12 & 24 & -8 \ 21 & 42 & -14 \end{array}\right]\].

Calculate (ii) Product D for (e)

The product \(D=BA\) is a scalar, calculated as:\[D=B A=3(-2)+6(4)+(-2)(7)=0\].

Key concepts

Matrix Operations

Matrix operations, particularly multiplication, are a cornerstone in mathematics, especially within linear algebra. When we talk about matrix multiplication, we mean the process of taking two matrices and producing a new matrix by following specific rules. This operation is not as straightforward as multiplying numbers or vectors.

The operation involves an element-by-element multiplication, but you must understand the crucial rule for compatibility. Only matrices whose dimensions match the criteria can be multiplied. Specifically, for two matrices, say, matrix \(A\) and \(B\), to be multiplied, the number of columns in \(A\) has to equal the number of rows in \(B\). This is what makes matrix multiplication valid.

• If matrix \(A\) is of size \(m \times n\), matrix \(B\) must be of size \(n \times p\).
• The resulting matrix \(C\), which is the product of \(A\) and \(B\), will be of size \(m \times p\).

Matrix multiplication is extensively used in various applications, from solving systems of equations to transformations in computer graphics. Understanding the fundamental process of matrix operations paves the way for exploring deeper mathematical territories.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and operations on vectors in these spaces, including matrices. Matrix multiplication is a fundamental concept within this field.

Linear algebra provides the framework and language to describe linear systems. Matrices can be thought of as collections of vectors laid out in a rectangular array and these matrices can represent systems of linear equations. Through matrix multiplication, one can perform many operations that facilitate solving these systems.

• Matrices serve as tools for mapping, or transforming, data from one vector space to another.
• The rules of matrix multiplication align well with the algebra of linear equations, bridging the gap between abstract mathematical theory and practical computational problems.

In linear algebra, concepts such as eigenvectors, eigenvalues, and matrix inverses rely heavily on the understanding of matrix multiplication. Thus, mastering this concept is essential for more advanced studies in the field of applied mathematics and engineering.

Matrix Dimensions

Matrix dimensions specify the size of a matrix, represented as \(m \times n\), where \(m\) is the number of rows, and \(n\) is the number of columns. Understanding matrix dimensions is imperative when performing any matrix operation such as addition, subtraction, and multiplication.

The dimensions dictate the computational possibilities of a matrix. Here’s why they matter:

• The dimensions indicate the matrix's capability to interact with another matrix.
• For addition and subtraction of matrices, the matrices involved must have the exact same dimensions.
• For multiplication, however, as previously mentioned, it’s crucial that the number of columns in the first matrix equals the number of rows in the second matrix.

When reading or writing a matrix's dimensions, the row count comes first, followed by the column count, like reading across and then down. This convention helps avoid confusion and ensures precise communication of matrix operations. Awareness of these dimensions ensures correct execution of matrix operations in all mathematical workflows.