2.2 Equal Matrices - SPM Mathematics

2.2 Equal Matrices

(A) Determining whether two matrices are equal

1. Two matrices are equal if they have the same order and their corresponding elements are equal.

For example,

(\begin{matrix} a & b \\ c & d \end{matrix}) = (\begin{matrix} e & f \\ g & h \end{matrix})

Therefore, a = e, b = f, c = g and d = h.

Example 1:

Determine whether the following pairs of matrices are equal.

\begin{array}{l} (a) A = (\begin{matrix} 10 & 8 \\ - 3 & 1 \end{matrix}) and B = (\begin{matrix} 10 & 8 \\ - 3 & 1 \end{matrix}) \\ (b) P = (\begin{array}{l} 2 \\ - 4 \\ 10 \end{array}) and Q = (\begin{array}{l} 2 \\ - 3 \\ 10 \end{array}) \\ (c) M = (\begin{array}{l} 3 \\ 5 \end{array}) and N = (4 7) \end{array}

Solution:

(a) Equal

(b) Not equal, because -4 ≠ -3.

(c) Not equal, because the orders of the matrices are not equal.

(B) Solving problem involving equal matrices

1. When matrices are equal, elements whose values are unknown can be determined.

Example 2:

State the value of the unknowns in the following pairs of equal matrix.

(\begin{array}{l} 2 x \\ x + 2 y \end{array}) = (\begin{array}{l} - 8 \\ 10 \end{array})

Solution:

(\begin{array}{l} 2 x \\ x + 2 y \end{array}) = (\begin{array}{l} - 8 \\ 10 \end{array})

2x = -8
x = -4

x + 2y = 10
(-4) + 2y = 10
2y = 10 + 4
2y = 14
y = 7