IB Mathematics (SL)/Matrices

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Topic 4: Matrices

Introduction

Organization of a matrix

The numbers in the matrix are called entries or elements

The order of a matrix defines its shape. For example, 2×1 matrix. First digit defines number of rows, second digit defines number of columns.

Matrices of order n×1 with just one column, are called column matrices:

For example: $\left[\begin{array}{c}1\\ 5\\ 6\end{array}\right]$

Matrices of order 1×n with just one row, are called row matrices

For example: $\left[\begin{array}{ccc}1& 5& 6\end{array}\right]$

A zero matrix is a matrix with all entries equal to zero.

For example: $\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

An identity matrix is a square matrix with entries of 1 on the leading diagonal (from top left to bottom right) and entries of zero everywhere else.

For example: $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Adding and subtracting matrices

Two or more matrices of identical dimensions $m$ and $n$ can be added. Given m-by-n matrices A and B, their sum A + B is the m-by-n matrix computed by adding corresponding elements For example:

$\left[\begin{array}{ccc}1& 3& 2\\ 1& 0& 0\\ 1& 2& 2\end{array}\right]+\left[\begin{array}{ccc}0& 0& 5\\ 7& 5& 0\\ 2& 1& 1\end{array}\right]=\left[\begin{array}{ccc}1+0& 3+0& 2+5\\ 1+7& 0+5& 0+0\\ 1+2& 2+1& 2+1\end{array}\right]=\left[\begin{array}{ccc}1& 3& 7\\ 8& 5& 0\\ 3& 3& 3\end{array}\right]$

Multiplying a matrix by a number

To multiply a matrix by a number, multiply every element by the number. For example:

$2\cdot \left[\begin{array}{ccc}1& 8& -3\\ 4& -2& 5\end{array}\right]=\left[\begin{array}{ccc}2\cdot 1& 2\cdot 8& 2\cdot -3\\ 2\cdot 4& 2\cdot -2& 2\cdot 5\end{array}\right]=\left[\begin{array}{ccc}2& 16& -6\\ 8& -4& 10\end{array}\right]$

Matrix multiplication

In order for matrix multiplication to occur, the number of columns in the first matrix must equal the number of rows in the second matrix. Consider Matrix A and B below:

$\left[\begin{array}{ccc}1& 3& 1\\ 2& -1& 1\end{array}\right]\cdot \left[\begin{array}{cc}1& 1\\ 2& -1\\ 3& 1\end{array}\right]$

Thus, the 2x3 and 3x2 matrices can be multiplied either way in this example. For this example, a 2x2 matrices will result. Note that if the 3x2 was multiplied by the 2x3, a 3x3 matrice would result. Proceed by multiplying the 1st row of Matrix A with the first column of Matrix B and summing the results. This will be your first element of the 2x2. Multiply the first row of Matrix A with the first column of Matrix B. This will be the adjacent element of the 2x2. This process is shown below.

Continuing with Matrix A and B from above

$\left[\begin{array}{ccc}1& 3& 1\\ 2& -1& 1\end{array}\right]\cdot \left[\begin{array}{cc}1& 1\\ 2& -1\\ 3& 1\end{array}\right]=\left[\begin{array}{cc}1+6+3& 1+-3+1\\ 2+-2+3& 2+1+1\end{array}\right]=\left[\begin{array}{cc}10& -1\\ 3& 4\end{array}\right]$

The 2x3 matrix is the most complex hand calculation the IBO will require on any examination, with subsequent matrices (3x3, 3x4, 4x4) being done through use of a GDC.

2×2 matrices

Similar to the process shown above, only simpler. A 2x2 mulitplied by a 2x2 will always yield a 2x2 matrix. Consider Matrix A and B below:

$\left[\begin{array}{cc}4& 2\\ -1& 8\end{array}\right]\cdot \left[\begin{array}{cc}-1& 3\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}-4+4& 12+0\\ 1+16& -3+0\end{array}\right]=\left[\begin{array}{cc}0& 12\\ 17& -3\end{array}\right]$

3×3 matrices

It is not advised by the IBO for SL students to calculate by hand 3x3 matrices, as it is merely a complex extension of multiplying 2x2 matrices. Instead, the IBO recommends the use of a GDC (Graphing Display Calculator), and the Matrix function on it.