Introduction to Electric Circuits

7.2 Matrices 143

Multiplication of matrices

Multiplication of two matrices is only possible if the number of rows in one of
the matrices is equal to the number of columns in the other. If two matrices (A
and B) are multiplied to give a third matrix (C) then any element (Cm~) of C is
found by adding the products of all the elements in row m of A with the
corresponding elements in column n of B.

Example 7.3

If

A [: :] and [: ;]

obtain (1) AB, (2) BA.

Solution

1 Let C = AB. Then

c [i

~1_ [(3 x 9) + (2 • 6)

(5•215

(3x4)+(2• 26]

(5• • 51 27

2 Let D = BA. Then

~] = [(9 • 3) + (4 x 5)

(6 • 3) + (7 • 5)

9x2 + 4Xl ] = E47 22]

(6•215 53 19

Note that C :/: D so that AB 4= BA. The multiplication process is therefore not
commutative and care must be taken to multiply in the correct order.

The determinant of a matrix

The determinant of a matrix is a number which is obtained by subtracting the
sum of the products of the elements along the diagonals to the left from the sum
of the products of the elements along the diagonals to the right. The symbol for
the determinant is A and the elements are enclosed by vertical lines.

Example 7.4

Find the determinant of the matrix

[i :1