Introduction to Matrix Algebra

Definition of Matrix

A matrix is a rectangular arrangement of numbers in rows and columns, and enclosed in brackets, thus:

\[ \begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\end{bmatrix} \]

The numbers are called elements of the matrix. The above matrix would be said to have three rows (first row: $ a_{11} $ $ a_{12} $ $ a_{13} $ $ a_{14} $ , and so on) and four columns, and would be called a matrix of dimension $ 3 $ . The elements are referred to by two subscripts; thus, the element in row 2, column 3 is element $ a_{23} $ . A matrix may have any number of rows and any number of columns; a general matrix is said to have dimension $ m n $ , thus, m rows and n columns.

Matrix Notation

Matrices are referred to by capital letters, thus: A, and by doubly subscripted lowercase letters enclosed in parentheses, thus: $ (a_{ij}) $ . If necessary for clarity, the dimension of the matrix is specified as a subscript, thus: $ A_{m n} $ .

Special Matrices

A matrix consisting of only one row is called a row matrix. A matrix consisting of only one column is called a column matrix. A matrix with equal numbers of rows and columns is called a square matrix. For a square matrix of dimension $ n n $ , the elements $ a_{11}, a_{22}, , a_{nn} $ are called the main diagonal elements. A matrix with all elements equal to zero is called a zero matrix. A zero matrix of dimension $ m n $ is denoted by $ 0_{m n} $ , or, if the dimension is clear from the context, simply 0.

EXAMPLE 34.1 $ $ is a row matrix (dimension $ 1 $ ). $ \[\begin{bmatrix}2 \\ -3\end{bmatrix}\]

$ is a column matrix (dimension $ 2 $ ).

Examples of square matrices are [4], $ \[\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\] $ , and $ \[\begin{bmatrix}-3 & 5 & -4 \\ 2 & 2 & -4 \\ 0 & 9 & -4\end{bmatrix}\] \[\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]

$ is also a zero matrix of dimension $ 2 $ .

Matrix Equality

Two matrices are equal if and only if they have the same dimension and corresponding elements are equal, thus, given $ A = (a_{ij}) $ and $ B = (b_{ij}) $ , A = B if and only if the matrices have the same dimension and $ a_{ij} = b_{ij} $ for all i and j.

Matrix Addition

Given matrices of the same dimension $ m n $ , $ A = (a_{ij}) $ and $ B = (b_{ij}) $ , the matrix sum $ A + B $ is defined by $ A + B = (a_{ij} + b_{ij}) $ , that is, $ A + B $ is a matrix of dimension $ m n $ with each element being the sum of the corresponding elements of A and B. The sum of two matrices of different dimensions is not defined.

Additive Inverse and Subtraction

The additive inverse, or negative, of an $ m n $ matrix $ A = (a_{ij}) $ is the $ m n $ matrix $ -A = (-a_{ij}) $ . Subtraction of two matrices of the same dimension $ m n $ , $ A = (a_{ij}) $ and $ B = (b_{ij}) $ , is defined by $ A - B = (a_{ij} - b_{ij}) $ , that is, A - B is a matrix of dimension $ m n $ with each element being the difference of the corresponding elements of A and B.

Properties of Matrix Addition

Given $ m n $ matrices A, B, C, and O, the following laws can be shown to hold (O is a zero matrix):

1. COMMUTATIVE LAW: $ A + B = B + A $

2. ASSOCIATIVE LAW: $ A + (B + C) = (A + B) + C $

3. IDENTITY LAW: $ A + O = A $

4. ADDITIVE INVERSE LAW: $ A + (-A) = O $

Product of a Matrix and a Scalar

The product of a matrix and a scalar is defined as follows: Given an $ m n $ matrix $ A = (a_{ij}) $ and a scalar (real number) c, then $ cA = (ca_{ij}) $ , that is, cA is the $ m n $ matrix formed by multiplying each element of A by c. The following properties can be shown to hold (A and B both of dimension $ m n $ ):

\[ c(A+B)=cA+cB \]

\[ (c+d)A=cA+dA \]

\[ (cd)A=c(dA) \]

SOLVED PROBLEMS

34.1. State the dimension of the following matrices: $ A = \[\begin{bmatrix} 3 \\ 4 \end{bmatrix}\] $ ; $ B = \[\begin{bmatrix} 4 & 3 \\ 5 & -2 \\ 6 & 4 \end{bmatrix}\] $ ; $ C = \[\begin{bmatrix} 0 & 0 & -3 \\ 4 & 2 & 2 \end{bmatrix}\]

$ .

A has 2 rows and 1 column; it is a $ 2 $ matrix.

B has 3 rows and 2 columns; it is a $ 3 $ matrix.

C has 2 rows and 3 columns; it is a $ 2 $ matrix.

34.2. Given the matrices $ A = \[\begin{bmatrix} 5 & 0 \\ 2 & -3 \end{bmatrix}\] $ , $ B = \[\begin{bmatrix} 3 & -2 \\ -4 & 8 \end{bmatrix}\] $ , $ C = \[\begin{bmatrix} -3 & -2 & -3 \\ 4 & 0 & 2 \end{bmatrix}\]

$ , find

1. $ A + B $ ; (b) -C; (c) $ B + C $ ; (d) B - A.

\[ \begin{aligned}(a)A+B=\begin{bmatrix}{{{5}}}&{{{0}}} \\{{{2}}}&{{{-3}}}\end{bmatrix}+\begin{bmatrix}{{{3}}}&{{{-2}}} \\{{{-4}}}&{{{8}}}\end{bmatrix}=\begin{bmatrix}{{{5}}}&{{{+3}}}&{{{0+(-2)}}} \\{{{2+(-4)}}}&{{{(-3)+8}}}\end{bmatrix}=\begin{bmatrix}{{{8}}}&{{{-2}}} \\{{{-2}}}&{{{5}}}\end{bmatrix}\end{aligned} \]

\[ \begin{aligned}(b)\ -C=\ -\begin{bmatrix}{{{-3}}}&{{{-2}}}&{{{-3}}} \\{{{4}}}&{{{0}}}&{{{2}}}\end{bmatrix}=\begin{bmatrix}{{{3}}}&{{{2}}}&{{{3}}} \\{{{-4}}}&{{{0}}}&{{{-2}}}\end{bmatrix}\end{aligned} \]

3. Since \(B\) is a \(2\times2\) matrix and \(C\) is a \(2\times3\) matrix, \(B+C\) is not defined.

\[ \left(\mathrm{d}\right)B-A=\begin{bmatrix}{{{3}}}&{{{-2}}} \\{{{-4}}}&{{{8}}}\end{bmatrix}-\begin{bmatrix}{{{5}}}&{{{0}}} \\{{{2}}}&{{{-3}}}\end{bmatrix}=\begin{bmatrix}{{{3}}}&{{{-5}}}&{{{(-2)-0}}} \\{{{(-4)-2}}}&{{{8-(-3)}}}\end{bmatrix}=\begin{bmatrix}{{{-2}}}&{{{-2}}} \\{{{-6}}}&{{{11}}}\end{bmatrix} \]

34.3. Verify the commutative law for matrix addition: for any two $ m n $ matrices A and B, $ A + B = B + A $ .

Let \(A = (a_{ij})\) and \(B = (b_{ij})\). Since both \(A\) and \(B\) have dimension \(m \times n\), both \(A + B\) and \(B + A\) are defined and have dimension \(m \times n\). Then

\[ A+B=(a_{ij})+(b_{ij})=(a_{ij}+b_{ij})\qquad and\qquad B+A=(b_{ij})+(a_{ij})=(b_{ij}+a_{ij}) \]

Since for all i and j, $ a_{ij} + b_{ij} $ and $ b_{ij} + a_{ij} $ are real numbers, $ a_{ij} + b_{ij} = b_{ij} + a_{ij} $ , Hence $ A + B = B + A $ .

34.4. Verify the identity law for matrix addition: for any $ m n $ matrix A, $ A + 0_{m n} = A $ .

Let \(A = (a_{ij})\); by definition \(0_{m \times n}\) is an \(m \times n\) matrix with all entries equal to zero, that is, \(0_{m \times n} = (0)\). Then \(A + 0_{m \times n}\) is defined and has dimension \(m \times n\), hence

\[ A+0_{m\times n}=(a_{ij})+(0)=(a_{ij}+0)=(a_{ij})=A \]

34.5. Given the matrices $ A = \[\begin{bmatrix} -2 & 6 & 2 \\ 0 & -3 & 4 \end{bmatrix}\] $ , $ B = \[\begin{bmatrix} 3 & -2 \\ -4 & 8 \end{bmatrix}\] $ , $ C = \[\begin{bmatrix} -3 & -2 & -3 \\ 4 & 0 & 2 \end{bmatrix}\]

$ , find

\[ a)-2A;(b)0B;(c)5B+3A;(d)-3C+4A. \]

\[ \begin{aligned}(a)\ -2A\ =-2\begin{bmatrix}{{{-2}}}&{{{6}}}&{{{2}}} \\{{{0}}}&{{{-3}}}&{{{4}}}\end{bmatrix}=\begin{bmatrix}{{{(-2)(-2)}}}&{{{(-2)6}}}&{{{(-2)2}}} \\{{{(-2)0}}}&{{{(-2)(-3)}}}&{{{(-2)4}}}\end{bmatrix}=\begin{bmatrix}{{{4}}}&{{{-12}}}&{{{-4}}} \\{{{0}}}&{{{6}}}&{{{-8}}}\end{bmatrix}\end{aligned} \]

\[ \begin{aligned}(b)0\boldsymbol{B}=\boldsymbol{0}\begin{bmatrix}{{{3}}}&{{{-2}}} \\{{{-4}}}&{{{8}}}\end{bmatrix}=\begin{bmatrix}{{{0(3)}}}&{{{0(-2)}}} \\{{{0(-4)}}}&{{{0(8)}}}\end{bmatrix}=\begin{bmatrix}{{{0}}}&{{{0}}} \\{{{0}}}&{{{0}}}\end{bmatrix}\end{aligned} \]

3. Since 5B is a $ 2 $ matrix and 3A is a $ 2 $ matrix, $ 5B + 3A $ is not defined.

\[ \begin{aligned}(d)-3C+4A&=-3\begin{bmatrix}{{{-3}}}&{{{-2}}}&{{{-3}}} \\{{{4}}}&{{{0}}}&{{{2}}}\end{bmatrix}+4\begin{bmatrix}{{{-2}}}&{{{6}}}&{{{2}}} \\{{{0}}}&{{{-3}}}&{{{4}}}\end{bmatrix}=\begin{bmatrix}{{{9}}}&{{{6}}}&{{{9}}} \\{{{-12}}}&{{{0}}}&{{{-6}}}\end{bmatrix}+\begin{bmatrix}{{{-8}}}&{{{24}}}&{{{8}}} \\{{{0}}}&{{{-12}}}&{{{16}}}\end{bmatrix}\\&=\begin{bmatrix}{{{1}}}&{{{30}}}&{{{17}}} \\{{{-12}}}&{{{-12}}}&{{{10}}}\end{bmatrix}\end{aligned} \]

34.6. Verify: If both A and B are $ m n $ matrices, then for any scalar c, $ c(A + B) = cA + cB $ .

First note that $ A + B $ , $ c(A + B) $ , cA, cB, and hence $ cA + cB $ are all defined and of dimension $ m n $ .

Let $ A = (a_{ii}) $ and $ B = (b_{ii}) $ ; then

\[ c(A+B)=c((a_{ij})+(b_{ij}))=c((a_{ij}+b_{ij}))=(c(a_{ij}+b_{ij})) \]

where the innermost multiplication is the product of two real numbers, and

\[ cA+cB=c(a_{ij})+c(b_{ij})=(ca_{ij}+cb_{ij}) \]

But by the distributive law for real numbers, $ c(a_{ij} + b_{ij}) = ca_{ij} + cb_{ij} $ for any i and j. Hence

\[ c(A+B)=cA+cB \]

SUPPLEMENTARY PROBLEMS

\[ 34.7.\ Given A=\begin{bmatrix}{{{3}}}&{{{4}}}&{{{-2}}} \\{{{8}}}&{{{0}}}&{{{2}}} \\{{{1}}}&{{{1}}}&{{{-2}}}\end{bmatrix},B=\begin{bmatrix}{{{4}}}&{{{2}}} \\{{{4}}}&{{{2}}} \\{{{-4}}}&{{{-2}}}\end{bmatrix},C=\begin{bmatrix}{{{0}}}&{{{2}}}&{{{0}}} \\{{{-3}}}&{{{-4}}}&{{{2}}} \\{{{7}}}&{{{2}}}&{{{-1}}}\end{bmatrix},find \]

1. $ A + B $ ; (b) $ A + C $ ; (c) B - B; (d) 2C.

\[ \begin{bmatrix}{{{3}}}&{{{6}}}&{{{-2}}} \\{{{5}}}&{{{-4}}}&{{{4}}} \\{{{8}}}&{{{3}}}&{{{-3}}}\end{bmatrix};(\mathrm{c})\begin{bmatrix}{{{0}}}&{{{0}}} \\{{{0}}}&{{{0}}} \\{{{0}}}&{{{0}}}\end{bmatrix};(\mathrm{d})\begin{bmatrix}{{{0}}}&{{{4}}}&{{{0}}} \\{{{-6}}}&{{{-8}}}&{{{4}}} \\{{{14}}}&{{{4}}}&{{{-2}}}\end{bmatrix} \]

34.8. Given A, B, and C as in the previous problem, find (a) $ 3A + 2C $ ; (b) $ B $ ; (c) -A - 2C.

\[ Ans.\quad(a)\begin{bmatrix}{{{9}}}&{{{16}}}&{{{-6}}} \\{{{18}}}&{{{-8}}}&{{{10}}} \\{{{17}}}&{{{7}}}&{{{-8}}}\end{bmatrix};(b)\begin{bmatrix}{{{1}}}&{{{1/2}}} \\{{{1}}}&{{{1/2}}} \\{{{-1}}}&{{{-1/2}}}\end{bmatrix};(c)\begin{bmatrix}{{{-3}}}&{{{-8}}}&{{{2}}} \\{{{-2}}}&{{{8}}}&{{{-6}}} \\{{{-15}}}&{{{-5}}}&{{{4}}}\end{bmatrix} \]

34.9. Verify the associative law for matrix addition: for any three $ m n $ matrices A, B, and C,

\[ A + (B + C) = (A + B) + C. \]

34.10. Verify the additive inverse law for matrix addition: for any $ m n $ matrix $ A, A + (-A) = O_{m n} $

34.11. Verify: for any two scalars c and d and any matrix A, $ (c + d)A = cA + dA $ .

34.12. Verify: for any two scalars c and d and any matrix A, (cd)A = c(dA).

34.13. The transpose of an $ m n $ matrix A is a matrix $ A^{T} $ formed by interchanging rows and columns of A, that is, an $ n m $ matrix with the element in row j, column i being $ a_{ij} $ . Find the transposes of matrices

\[ \begin{aligned}(a)A=\begin{bmatrix}{{{3}}}&{{{4}}}&{{{-2}}} \\{{{8}}}&{{{0}}}&{{{2}}} \\{{{1}}}&{{{1}}}&{{{-2}}}\end{bmatrix};(b)B=\begin{bmatrix}{{{4}}}&{{{2}}} \\{{{4}}}&{{{2}}} \\{{{-4}}}&{{{-2}}}\end{bmatrix}.\end{aligned} \]

\[ Ans.\quad(a)A^{T}=\begin{bmatrix}{{{3}}}&{{{8}}}&{{{1}}} \\{{{4}}}&{{{0}}}&{{{1}}} \\{{{-2}}}&{{{2}}}&{{{-2}}}\end{bmatrix};(b)B^{T}=\begin{bmatrix}{{{4}}}&{{{4}}}&{{{-4}}} \\{{{2}}}&{{{2}}}&{{{-2}}}\end{bmatrix} \]

34.14. Verify: (a) $ (A^{}){} = A $ ; (b) $ (A + B)^{} = A^{} + B^{} $ ; (c) $ (cA)^{} = cA^{} $