MATRICES AND DETERMINANTS

CONTENT

• Multiplication of Matrices
• Transpose of a Matrix
• Determinant of 2 × 2 and 3 × 3 Matrices
• Application to Solving Simultaneous Linear Equations in Two Variables

Multiplication of Matrices

Let A and B be matrices. The product matrix AB exists if the number of columns of matrix A is the same as the number of the rows of matrixes B. A is then premultiplied by B. Where AB exists we say matrices A and B are conformable. In general matrices A of order m × n will premultiply matrix B of order n × p to give matrix C of order m  p. Notice the ns drop out.

Example 2:

Let A = \(\begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & 3\end{pmatrix}\) and B = \(\begin{pmatrix} 2 & 1\\ 0 & -2 \\ 3 & -1\end{pmatrix}\) 

AB = \(\begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & 3\end{pmatrix}\) \(\begin{pmatrix} 2 & 1\\ 0 & -2 \\ 3 & -1\end{pmatrix}\) 

= \(\begin{pmatrix} 1×2 &+  & 2×0 &+ & -1×3 \qquad 1×1 &+& 2×-2 &+& -1×-1\\ 0×2 &+ & 1×0 &+ & 3×3 \qquad 0×1 &+& 1×-2 &+& 3×-1\end{pmatrix}\) = \(\begin{pmatrix} -1 & -2\\ 9 & -5\end{pmatrix}\)

But BA = \(\begin{pmatrix} 2 & 1\\ 0 & -2 \\ 3 & -1\end{pmatrix}\) \(\begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & 3\end{pmatrix}\)

= \(\begin{pmatrix} 2×1 &+  & 1×0 \qquad 2 × 2 &+ & 1×1 \qquad 2×-1 &+& 1×3\\ 0×1 &+ & -2(0) \qquad 0×2 &+ & -2×1 \qquad 0×-1 &+& -2×3\\ 3×1 &+ & 0×-1 \qquad 3×2 &+ & -1×1 \qquad 3×-1 &+& -1×3\end{pmatrix}\) = \(\begin{pmatrix} 2 & 5 & 1\\ 0 & -2 & -6\\ 3 & 5 &-6\end{pmatrix}\)

In general, matrix multiplication is not commutative as AB ≠ BA.