Equality of two Matrices

Two matrices A = [a_ij] and B = [b_ij] are said to be equal if they are of same order and each element of A is equal to the corresponding element of B, that is a_ij = b_ij for all i and j.  Symbolically we write it as A = B

Find the values of a, b, c, x, y and z
Solution: Since the matrices are equal, corresponding elements are equal
x + 3 = 0
x = -3         
z + 4 = 6  
z = 2              
2y – 7 = 3y - 2   
2y - 3y = -2 + 7
y = -5                                            
a – 1 = -3
a = -2              
2c + 2 = 0
c = -1                                  
b – 3 = 2b + 4
b - 2b = 7
b = -7
Hence, a = -2, b = -7, c = -1, x = -3, y = -5 and z = 2.
Try this:
1. Given that the following matrices are equal, find the values of x and y.

(Answer: x = 1, y = 4)
2. Given that the following matrices are equal, find the values of x, y, and z.

(Answer: x = 4, y = –6, and z = 9)

Construction of a Matrix

When the general term and the order of a matrix is given, we can easily construct a matrix.
For example:  Construct a 3 x 4 matrix whose elements are given by a_ij = 2i - j

a11 = 2 – 1 = 1    a₁₂ = 2 – 2 = 0    a₁₃ = 2 – 3 = -1    a₁₄ = 2 – 4 = -2
a₂₁ = 4 – 1 = 3    a₂₂ = 4 – 2 = 2    a₂₃ = 4 – 3 = 1      a₂₄ = 4 – 4 = 0
a₃₁ = 6 - 1 = 5     a₃₂ = 6 – 2 = 4    a₃₃ = 6 – 3 = 3      a₃₄ = 6 – 4 = 2

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Reference Links:

• http://en.wikipedia.org/wiki/Matrix_(mathematics)
• http://www.mathreference.com/la-mpoly,order.html
• http://wiki.answers.com/Q/What_is_order_of_the_resultant_matrix_AB_when_two_matrices_are_multiplied_and_the_order_of_the_Matrix_A_is_m_n_order_of_Matrix_B_is_n_p_Also_state_the_condition_under_which_two_matrices_can_be_mult