How do you find the determinant and inverse of a 2x2 matrix?

For A = [[a,b],[c,d]], det A = ad - bc. If det A is not zero, the inverse is A^(-1) = (1/det A) times [[d,-b],[-c,a]]. If det A = 0, A has no inverse.

With T = 0.7 & 0.4 \ 0.3 & 0.6 and initial state 600\\400, find the distribution after 1 week.

T600\\400 = 0.7(600)+0.4(400)\\0.3(600)+0.6(400) = 580\\420. GDC: Compute T directly in matrix mode.

Find AB where A = 1 & 0 & 2\\0 & 1 & 0\\3 & 0 & 1 and B = 1 & 2\\0 & 1\\4 & 0.

AB is 3 x 2. Computing row by row: Row 1: (1(1)+0(0)+2(4), 1(2)+0(1)+2(0)) = (9, 2). Row 2: (0+0+0, 0+1+0) = (0, 1). Row 3: (3+0+4, 6+0+0) = (7, 6). AB = 9&2\\0&1\\7&6. GDC: Enter A as a 3 x 3 matrix and B as 3 x 2, then compute [A][B].

Matrix Operations (AI HL)

A matrix is just a rectangular array of numbers, but the rules for combining two of them are stricter than ordinary arithmetic - addition needs matching dimensions, and multiplication needs the first matrix's columns to match the second's rows. This page covers those operations, plus the \(2\times2\) determinant and inverse, with worked examples and the errors that cost marks. It's part of the broader Matrices topic.

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Determinant, inverse and multiplication

Covered under IB syllabus reference AHL1.14: the algebra of matrices (equality, addition, subtraction, scalar multiplication), multiplication of matrices and its properties, and determinants and inverses of \(2\times2\) matrices by hand.

Determinant of a 2×2 matrix

\(\det A = ad-bc\)

For \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). In the formula booklet - a zero determinant means \(A\) has no inverse.

Inverse of a 2×2 matrix

\(A^{-1}=\dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)

Also in the formula booklet. For larger matrices, find the inverse with your GDC.

Matrix multiplication

Only defined when the number of columns in the first matrix equals the number of rows in the second. Associative and distributive, but \(AB\neq BA\) in general - the order you multiply in matters.

Not in the formula booklet - definition

Need the full syllabus wording and formula-booklet reference table? See Matrices.

Worked examples

Medium

GDC

[3 marks]

With \(T = \begin{pmatrix} 0.7 & 0.4 \\ 0.3 & 0.6 \end{pmatrix}\) and initial state \(\begin{pmatrix}600\\400\end{pmatrix}\), find the distribution after 1 week.

Worked solution

\(T\begin{pmatrix}600\\400\end{pmatrix} = \begin{pmatrix}0.7(600)+0.4(400)\\0.3(600)+0.6(400)\end{pmatrix}\) M1
\(= \begin{pmatrix}580\\420\end{pmatrix}.\) A1 A1

GDC: Compute \(T\mathbf{v}\) directly in matrix mode.

M1 Multiply A1 Correct value A: 580 A1 Correct value B: 420

Hard

GDC

[4 marks]

Find \(AB\) where \(A = \begin{pmatrix}1 & 0 & 2\\0 & 1 & 0\\3 & 0 & 1\end{pmatrix}\) and \(B = \begin{pmatrix}1 & 2\\0 & 1\\4 & 0\end{pmatrix}.\)

Worked solution

\(AB\) is \(3\times2\). Computing row by row: M1
Row 1: \((1(1)+0(0)+2(4),\ 1(2)+0(1)+2(0)) = (9,\ 2).\) Row 2: \((0+0+0,\ 0+1+0) = (0,\ 1).\) Row 3: \((3+0+4,\ 6+0+0)\) A1
\(= (7,\ 6).\) A1
\(AB = \begin{pmatrix}9&2\\0&1\\7&6\end{pmatrix}.\) A1

GDC: Enter \(A\) as a \(3\times3\) matrix and \(B\) as \(3\times2\), then compute \([A][B]\).

M1 Row-by-column method A1 Rows 1 and 2 A1 Row 3 A1 Complete answer

Common mistakes

Assuming matrix multiplication is commutative. \(AB\) and \(BA\) are generally different matrices, and one might not even be defined - never swap the order without checking the dimensions line up.
Multiplying matrices whose dimensions don't allow it. \(AB\) only exists when the number of columns in \(A\) equals the number of rows in \(B\) - check this before starting the row-by-column arithmetic, not after.
Forgetting a zero determinant means no inverse. If \(\det A=0\), the matrix is singular and \(A^{-1}\) doesn't exist - though the IB guarantees \(A\) is invertible whenever a question asks you to find \(A^{-1}\).

Full topicMatrices RelatedEigenvalues and eigenvectors FormulasAI HL formula booklet explained PrintAI HL formula sheet

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Quick answers

How do you multiply two matrices together?

Multiply each row of the first matrix by each column of the second, summing the products, so entry \((i,j)\) of the result is row \(i\) of \(A\) dotted with column \(j\) of \(B\). This only works when the columns of \(A\) match the rows of \(B\).

How do you find the determinant and inverse of a 2×2 matrix?

For \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), \(\det A = ad-bc\). If \(\det A \neq 0\), the inverse is \(A^{-1}=\tfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\). Also check using your GDC for larger matrices.

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