Algebra of Matrices of Order 2 and 3

EQUALITY OF MATRICES

DEFINITION: 

Two matrices A = [a_ij] _mxn And B = [b_ij] _mxn are said to be equal if

1. m=r, i.e., the number of rows in A equals the number of rows in B
2. n=s, i.e., the number of columns in A equals the number of columns in B
3. a_ij=b_ij  for i=1,2,..., in and j=1,2,....,n 

If two matrices A and B are equal, we write A = B, otherwise we write A # B.

ILLUSTRATION 1:

The matrices are equal if x=-1, y=0 and z=4.

ILLUSTRATION 2:

If , find x, y, z, w.

SOLUTION

Since the corresponding elements of two equal matrices are equal.

Therefore,

 

x — y = —1, 2x -V z = 5, 2x — y = 0, 3z w = 13.

Solving the equation x-y=-1 and 2x-y=0 as simultaneous linear equations, we get x=1,y=2.

Now, putting x=1 in 2x+z=5, we get z=3.

Substituting z=3 in 3z+w=13, we obtain w=4.

Thus, the given matrices are equal if x=1, y=2, z=3 and w=4.

ILLUSTRATION 3:

Find the values of x,y,z an a which satisfy the matrix equation.

PROOF:

Since the corresponding elements of two equal matrices are equal,

Therefore,

 x + 3 = 0, 2y x = —7, z — 1 = 3 and 4a - 6 = 2a.

Solving these equations, we get

a = 3, x = —3, y = —2, z = 4.

ADDITION OF MATRICES

DEFINITION: 

Let A, B be two matrices, each of order m x n. Then their sum A+B is a matrix of order m x n and is obtained by adding the corresponding element of A and B.

Thus, if A = [a_ij] _mxn and B = [b_ij] _mxn are two matrices of the same order, their sum A+B is defined to be the matrix of order m x n such that (A+ B) _ij = + bij for i =1, 2,…, m and j =1, 2,…, n

NOTE: The sum of two matrices is defined only when they are of the same order.

ILLUSTRATION 1:

ILLUSTRATION 2:  If ’, then

A+B is not defined, because A and B are not of the same order.

ILLUSTRATION 3: For the following pairs of matrices A-FB is not defined because they are of different orders:

i. 

ii. 

PROPERTIES OF MATRIX ADDITION

i. Matrix addition is commutative

i.e., if A and B are two m x n matrices, then A+B=B+A.

Existence of Inverse

For every matrix A = [a_ij] _mxn there exists a matrix [—a_ij] _mxn, denoted by -A, such that A+ (-A) = 0 = (-A) + A.

MULTIPLICATION OF A MATRIX BY A SCALAR

DEFINITION 

Let A = [a_ij] be an m x n matrix and k be any number called a scalar. Then the matrix obtained by multiplying every element of A by k is called the scalar multiple of A by k and is denoted by kA.

Thus, kA = [ka_ij] _mxn.

SUBTRACTION OF MATRICES

DEFINITION 

For two matrices A and B of the same order, we define A—B=A + (—B).

ILLUSTRATION 2:

If , find 3A — 2B.

SOLUTION

We have.

3A-2B= 3A + (-2) B

MULTIPLICATION OF MATRICES

Two matrices A and B are conformable for the product AB if the number of column in A(pre-multiplier) is same as the number of tows in B (post-multiplier).

ILLUSTRATION 2 Let 

find AB and BA and show that AB ≠ BA.

SOLUTION

Here, A is a 2 x 3 matrix and B is a 3 x 2 matrix. So, AB exists and it is of order 2 x 2.

Again, B is a 3 x 2 matrix and A is a 2 x 3 matrix. So, BA exists and it is of order 3 x 3.

Now,

Hence, AB ≠ BA.

PROPERTIES OF MATRIX MULTIPLICATION

• Matrix multiplication is not commutative in general.
• Matrix multiplication is associative i.e. (AB) C=A (BC), whenever both sides are defined.
• Matrix multiplication is distributive over matrix addition i.e.,
• A (B + C) = AB + AC, 
• (A + B) C = AB + AC,
  whenever both sides of equality are defined.
• If A is an m x n matrix, then I_m A = A = A I_n
• The product of two matrices can be the null matrix while neither of them is the null matrix. For example, if

      while neither A nor B is the null matrix

• If A is m x n matrix and 0 is a null matrix, then
•  A_mxnO_nxp = O_mxp
• O_pxmA_mxn = O_pxn

TRANSPOE OF A MATRIX

DEFINITION 

Let A= [a_ij] be an m x n matrix. Then the transpose of A, denoted by A^T or A', is an n x m matrix such that

(A^T )_ij= a_ij for all i=1,2,...,m’ j=1,2,...,n.

Thus, A^T is obtained from A by changing its rows into columns and its columns into rows. For example, if

 

PROPERTIES OF TRANSPOSE

Let A and B be two matrices. Then.

• (A^T) ^T = A
• (A + B) ^T = A^T + B^T, A and B being of the same order.
• (KA) ^T = kA^T, k be any scalar (real or complex)
• (AB) ^T = B^TA^T, A and B being conformable for the product AB.