Operations on Matrices

DEFINITION 1.2.1 (Transpose of a Matrix)   The transpose of an $m \times n$ matrix $A=[a_{ij}]$ is defined as the $n \times m$ matrix $B = [b_{ij}],$ with $b_{ij} = a_{ji}$ for $1 \leq i \leq m$ and $1 \leq j \leq n.$ The transpose of $A$ is denoted by $A^t.$

That is, by the transpose of an $m \times n$ matrix $A,$ we mean a matrix of order $n \times m$ having the rows of $A$ as its columns and the columns of $A$ as its rows.

For example, if $A=\begin{bmatrix}1 & 4 & 5 \\ 0 & 1 & 2 \end{bmatrix}$ then $A^t =\begin{bmatrix}1 & 0 \\ 4 & 1 \\ 5 & 2 \end{bmatrix}.$

Thus, the transpose of a row vector is a column vector and vice-versa.

THEOREM 1.2.2   For any matrix $A, \;$ $\; (A^t)^t = A.$

Proof. Let $A = [a_{ij}], \; A^t = [b_{ij}]$ and $(A^t)^t = [c_{ij}].$ Then, the definition of transpose gives

$$c_{ij} = b_{ji} = a_{ij} \;\; {\mbox{ for all }} \;\; i,j$$

and the result follows. height6pt width 6pt depth 0pt

DEFINITION 1.2.3 (Addition of Matrices)   let $A=[a_{ij}]$ and $B= [b_{ij}]$ be are two $m \times n$ matrices. Then the sum $A + B$ is defined to be the matrix $C = [c_{ij}]$ with $c_{ij} = a_{ij} + b_{ij}.$

Note that, we define the sum of two matrices only when the order of the two matrices are same.

DEFINITION 1.2.4 (Multiplying a Scalar to a Matrix)   Let $A=[a_{ij}]$ be an $m \times n$ matrix. Then for any element $k \in {\mathbb{R}},$ we define $k A = [k a_{ij}].$

For example, if $A=\begin{bmatrix}1 & 4 & 5 \\ 0 & 1 & 2 \end{bmatrix}$ and $k = 5,$ then $5 A = \begin{bmatrix}5 & 20 & 25 \\ 0 & 5 & 10 \end{bmatrix}.$

THEOREM 1.2.5   Let $A, B$ and $C$ be matrices of order $m \times n,$ and let $k, \ell \in {\mathbb{R}}.$ Then

1. $A + B = B + A \; \hspace{1.5in} {\mbox{ (commutativity)}}.$
2. $( A + B ) + C = A + (B + C) \; \hspace{0.635in} {\mbox{ (associativity)}}.$
3. $k ( \ell A) = (k \ell) A.$
4. $( k+ \ell) A = k A + \ell A.$

Proof. Part 1.
Let $A=[a_{ij}]$ and $B= [b_{ij}].$ Then

$$A + B = [a_{ij}] + [b_{ij}] = [ a_{ij} + b_{ij} ] = [ b_{ij} + a_{ij}] = [b_{ij}] + [a_{ij}]= B + A$$

as real numbers commute.

The reader is required to prove the other parts as all the results follow from the properties of real numbers. height6pt width 6pt depth 0pt

EXERCISE 1.2.6  

1. Suppose $A + B = A.$ Then show that $B = {\mathbf 0}.$
2. Suppose $A + B = {\mathbf 0}.$ Then show that $B = (-1) A = [- a_{ij}].$

DEFINITION 1.2.7 (Additive Inverse)   Let $A$ be an $m \times n$ matrix.

1. Then there exists a matrix $B$ with $A + B = {\mathbf 0}.$ This matrix $B$ is called the additive inverse of $A,$ and is denoted by $- A = (-1) A.$
2. Also, for the matrix ${\mathbf 0}_{m \times n},$ $A + {\mathbf 0}= {\mathbf 0}+ A = A.$ Hence, the matrix ${\mathbf 0}_{m \times n}$ is called the additive identity.