Working with 2 x 2 Matrices

Matrices are a useful way to represent systems of linear equations, especially for large numbers of equations and variables.

For example, these equations:

\(5x - 2y = 7\)
\(2x + y = 10\)

can be represented with matrices as

\[\begin{bmatrix} 5 & 2 \\ 2 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 7 \\ 10 \\ \end{bmatrix}\]

A general representation of a system with matrices is Ax = b, where the bold type denotes that A, x, and b represent matrices.

Defined Operations for Matrices

A few operations are defined for matrices: addition, subtraction, multiplication, the inverse, and the determinant. Note that there is no division defined for matrices and matrix multiplication is not commutative. Also, not all matrices can be multiplied: the number of columns in the first matrix must equal the number of rows in the second matrix.

Operations beyond addition and subtraction usually require technology such as computers or advanced calculators. However, because of their simplicity, 2x2 matrices have some shortcuts that are easier to compute by hand or with a simple calculator.

Multiplication

For multiplication, multiply the corresponding rows in the first matrix with corresponding columns in the second matrix to get the entry in the resulting matrix:

\[\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} e & f \\ g & h \\ \end{bmatrix} = \begin{bmatrix} (ae+bg) & (af+bh) \\ (ce+dg) & (cf+dh) \\ \end{bmatrix}\]

Multiplication was defined this way because it turns out to have many applications, such as finding geometric transformations.

Finding the Determinant

The determinant is a property of a matrix used to find the inverse of the matrix and hence the solution of the system of equations. The determinant is represented with \(\vert \vert\) instead of [ ].

For 2x2 matrices:

\[\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = ad - cb\]

Note that this does not extend to larger matrices.

In our example the determinant is (5)(1) - (2)(2) = 5 - 4 = 1

Finding the Inverse

The inverse of a matrix is defined so that the inverse multiplied by the original matrix = 1, or:

\[\boldsymbol{AA}^{-1} = 1\]

For a 2x2 matrix, the inverse of:

\(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\) is:

\[\frac{1}{\begin{vmatrix} a & b \\ c & d \\ \end{vmatrix}} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\]

The matrix on the bottom is the adjoint of A and a 2x2 matrix is formed by switching the positions of the a and d entries and taking the opposite of the b and c entries. Note that if the determinant is zero then the matrix cannot be inverted.

In our example, the inverse is \(\frac{1}{1}\begin{bmatrix} 1 & -2 \\ -2 & 5 \\ \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ -2 & 5 \\ \end{bmatrix}\)

Finding the Solution

The solution of the given system is

\[\boldsymbol{x} = \boldsymbol{A}^{-1} \boldsymbol{b}\]

In our example \(\begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ -2 & 5 \\ \end{bmatrix} \begin{bmatrix} 7 \\ 10 \\ \end{bmatrix}\)

So,

\[x = (1)(7) + (-2)(10) = 7 - 20 = -13\]

and

\[y = (-2)(7) + (5)(10) = -36\]