Identity, determinant and inverse of a 2 × 2 matrix

Warm-up

Three quick questions from earlier lessons.

Discover

The number 5 has the inverse 1/5 because 5 · 1/5 = 1. Which number has no inverse?

Key idea

A 2 × 2 matrix has an inverse exactly when its determinant is not 0.

The identity

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

plays the role of 1 (

AI = IA = A

) and the zero matrix the role of 0. For

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

\det A = ad - bc

. If

\det A \ne 0

A^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

and

AA^{-1} = I

; if

\det A = 0

A

has no inverse.

Worked example

Invert $\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$ .

4 · 6 − 7 · 2 = 10

Determinant.

\begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix}

Swap and change signs.

\begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix}

Divide by 10.

Your turn, guided

Your turn: det $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$ .

1

3 · 2 − 1 · 5.

Practice ★

What is the determinant of $\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}$ ?
What is the determinant of $\begin{pmatrix} 7 & 3 \\ 5 & 4 \end{pmatrix}$ ?

For which value of k does $\begin{pmatrix} k & 3 \\ 10 & 2 \end{pmatrix}$ have no inverse?

Watch out!

Practice ★★

Let A = $\begin{pmatrix} 5 & 3 \\ 1 & 8 \end{pmatrix}$ . What is the entry in row 1, column 1 of the inverse of A?
Let A = $\begin{pmatrix} -5 & 2 \\ 7 & 4 \end{pmatrix}$ . What is the entry in row 1, column 1 of the inverse of A?

What is the determinant of $\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}$ ?

Exit ticket

Three questions to check you have got it.

What is the determinant of $\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}$ ?
What is the determinant of $\begin{pmatrix} 7 & 3 \\ 5 & 4 \end{pmatrix}$ ?
What is the determinant of $\begin{pmatrix} 7 & 8 \\ 3 & 7 \end{pmatrix}$ ?

Summary card

Determinants and inverses

det = ad − bc; inverse = (1/det)(d, −b; −c, a) when det ≠ 0.

Watch out: the inverse is not the matrix of reciprocals.

End of the lesson

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