What does it mean for a matrix to be singular?

Its determinant is 0, so it has no inverse.

What is a non-singular matrix?

One with non-zero determinant; it has an inverse.

Inverse of \begin{pmatrix}a&b\\ c&d\end{pmatrix}?

\frac{1}{ad-bc}\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}

How do you form the adjugate of a 2\times2 matrix?

Swap the leading-diagonal entries and change the signs of the other two.

Inverse of \begin{pmatrix}4&3\\ 2&2\end{pmatrix}?

\frac12\begin{pmatrix}2&-3\\ -2&4\end{pmatrix}

Solve \mathbf{AX}=\mathbf B for \mathbf X.

\mathbf X=\mathbf A^{-1}\mathbf B (pre-multiply).

Solve \mathbf{XA}=\mathbf B for \mathbf X.

\mathbf X=\mathbf B\mathbf A^{-1} (post-multiply).

State the inverse of a product.

(\mathbf{AB})^{-1}=\mathbf B^{-1}\mathbf A^{-1}

Does a matrix with \det=-6 have an inverse?

Yes: only \det=0 means no inverse.

Determinants and inverses of 2x2 matricesEdexcel International A Level Further Maths: Flashcards

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Determinant of $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$?

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Determinant of $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ ?: $ad-bc$
What does it mean for a matrix to be singular?: Its determinant is $0$ , so it has no inverse.
What is a non-singular matrix?: One with non-zero determinant; it has an inverse.
Inverse of $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ ?: $\frac{1}{ad-bc}\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}$
How do you form the adjugate of a $2\times2$ matrix?: Swap the leading-diagonal entries and change the signs of the other two.
$\det\begin{pmatrix}4&3\\ 2&2\end{pmatrix}$ ?: $4(2)-3(2)=2$
Inverse of $\begin{pmatrix}4&3\\ 2&2\end{pmatrix}$ ?: $\frac12\begin{pmatrix}2&-3\\ -2&4\end{pmatrix}$
What does $\mathbf{AA}^{-1}$ equal?: The identity matrix $\mathbf I$ .
Solve $\mathbf{AX}=\mathbf B$ for $\mathbf X$ .: $\mathbf X=\mathbf A^{-1}\mathbf B$ (pre-multiply).
Solve $\mathbf{XA}=\mathbf B$ for $\mathbf X$ .: $\mathbf X=\mathbf B\mathbf A^{-1}$ (post-multiply).
State the inverse of a product.: $(\mathbf{AB})^{-1}=\mathbf B^{-1}\mathbf A^{-1}$
For which $k$ is $\begin{pmatrix}k&4\\ 3&k+1\end{pmatrix}$ singular?: $k^2+k-12=0$ , so $k=3$ or $k=-4$ .
Does a matrix with $\det=-6$ have an inverse?: Yes: only $\det=0$ means no inverse.

Exam questions on Determinants and inverses of 2x2 matrices

$\mathbf{A}=\begin{pmatrix}4&3\\ 2&2\end{pmatrix}$ .
Given that $\mathbf{AX}=\begin{pmatrix}2&0\\ 1&1\end{pmatrix}$ , find the matrix $\mathbf{X}$ .2 marks
$\mathbf{B}=\begin{pmatrix}k&4\\ 3&k+1\end{pmatrix}$ , where $k$ is a constant.
Given that $k=2$ , find $\mathbf{B}^{-1}$ .2 marks
$\mathbf{C}=\begin{pmatrix}2&1\\ 5&3\end{pmatrix}$ and $\mathbf{D}=\begin{pmatrix}1&2\\ 1&3\end{pmatrix}$ .
Find $\mathbf{CD}$ and hence find $(\mathbf{CD})^{-1}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).