The minor multiplied by the sign from the pattern +\,-\,+;\;-\,+\,-;\;+\,-\,+.

How do you find \det\mathbf{A} for a 3\times3 matrix?

Expand along any row or column: sum of entry \times cofactor.

What does \det\mathbf{M}=0 mean?

The matrix is singular and has no inverse.

What does a negative determinant tell you?

The transformation reverses orientation.

What does a positive determinant tell you?

The transformation preserves orientation.

Steps to find \mathbf{A}^{-1} for a 3\times3 matrix?

Find \det\mathbf{A}, the matrix of cofactors, transpose it, then divide by \det\mathbf{A}.

What is the adjugate of \mathbf{A}?

The transpose of the matrix of cofactors.

Which cofactor gives the entry in row 3, column 2 of \mathbf{A}^{-1}?

The cofactor of the entry in row 2, column 3 of \mathbf{A}, divided by \det\mathbf{A}.

3x3 determinants and inverse matricesAQA A-Level Further Maths: Flashcards

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What is a minor of a $3\times3$ matrix?

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What is a minor of a $3\times3$ matrix?: The determinant of the $2\times2$ matrix left after deleting the entry's row and column.
What is a cofactor?: The minor multiplied by the sign from the pattern $+\,-\,+;\;-\,+\,-;\;+\,-\,+$ .
How do you find $\det\mathbf{A}$ for a $3\times3$ matrix?: Expand along any row or column: sum of entry $\times$ cofactor.
What does $\det\mathbf{M}=0$ mean?: The matrix is singular and has no inverse.
What is the area scale factor of a $2\times2$ transformation?: $|\det\mathbf{M}|$
What is the volume scale factor of a $3\times3$ transformation?: $|\det\mathbf{M}|$
What does a negative determinant tell you?: The transformation reverses orientation.
What does a positive determinant tell you?: The transformation preserves orientation.
Steps to find $\mathbf{A}^{-1}$ for a $3\times3$ matrix?: Find $\det\mathbf{A}$ , the matrix of cofactors, transpose it, then divide by $\det\mathbf{A}$ .
What is the adjugate of $\mathbf{A}$ ?: The transpose of the matrix of cofactors.
Which cofactor gives the entry in row 3, column 2 of $\mathbf{A}^{-1}$ ?: The cofactor of the entry in row 2, column 3 of $\mathbf{A}$ , divided by $\det\mathbf{A}$ .
If the volume scale factor of $\mathbf{M}$ is $7$ and a solid has volume $4$ , what is the image volume?: $4\times7=28$

Exam questions on 3x3 determinants and inverse matrices

The matrix $\mathbf{A}=\begin{pmatrix}1&2&0\\3&1&1\\0&2&1\end{pmatrix}$ .
State whether $\mathbf{A}$ preserves or reverses orientation, giving a reason.2 marks
The matrix $\mathbf{B}=\begin{pmatrix}2&0&1\\1&1&0\\0&3&1\end{pmatrix}$ , for which $\det\mathbf{B}=5$ .
Find the entry in row 3, column 2 of $\mathbf{B}^{-1}$ .2 marks
The matrix $\mathbf{C}=\begin{pmatrix}1&1&1\\1&2&3\\1&4&9\end{pmatrix}$ .
Show that $\det\mathbf{C}=2$ .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).