Determinants and Inverses (AI HL)
The determinant of a matrix is a single number that tells you whether it can be inverted, and the inverse itself lets you "undo" a matrix in calculations such as solving a system of equations. This page covers finding both by hand for \(2\times2\) matrices, with worked examples and the mistakes that lose the most marks. It's part of the broader Matrices topic.
16 questions on this sub-topic.
The determinant and inverse formulas
Covered under IB syllabus reference AHL1.14: determinants and inverses of \(n\times n\) matrices with technology, and by hand for \(2\times2\) matrices. Both formulas below are given in the formula booklet.
Determinant of a \(2\times2\) matrix
\(\det M = ad - bc\) for \(M = \begin{pmatrix}a&b\\c&d\end{pmatrix}\)
A determinant of \(0\) means the matrix is singular - it has no inverse, and any system it represents has no unique solution.
Inverse of a \(2\times2\) matrix
\(M^{-1} = \dfrac{1}{\det M}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)
Swap the leading-diagonal entries, negate the other two, then divide everything by the determinant. Only defined when \(\det M \neq 0\).
Need the full syllabus wording and formula-booklet reference table? See Matrices. GDC key sequences for matrix mode live in the Matrices GDC section.
Worked examples
Show that \(M = \begin{pmatrix} 2 & 6 \\ 1 & 3 \end{pmatrix}\) has no inverse.
Worked solution
\(\det M = (2)(3) - (6)(1)\) M1
\(= 0.\) A1
Since \(\det = 0\), \(M\) is singular and has no inverse. A1 AG
A shop sells apples and bananas. 3 apples and 2 bananas cost £2.80; 1 apple and 4 bananas cost £2.20. Set up and solve the system using matrix inverses, giving prices in pence.
Worked solution
Let \(a\) = price of apple, \(b\) = price of banana (pence). System: \(\begin{pmatrix}3&2\\1&4\end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}280\\220\end{pmatrix}.\) M1
\(\det = 12-2=10.\) \(A^{-1} = \tfrac{1}{10}\begin{pmatrix}4&-2\\-1&3\end{pmatrix}.\) A1
\(\begin{pmatrix}a\\b\end{pmatrix} = \tfrac{1}{10}\begin{pmatrix}4(280)-2(220)\\-280+3(220)\end{pmatrix}\) M1
\(= \tfrac{1}{10}\begin{pmatrix}680\\380\end{pmatrix}.\) A1
\(a = 68\) p, \(b = 38\) p. A1
\(P = \begin{pmatrix}t+1 & 2 \\ 3 & t-1\end{pmatrix}\) is singular.
(a)(i) Find the value of \(t<0\).
(a)(ii) Find the value of \(t>0.\)
Worked solution
(a)(i) \(t = -\sqrt{7}\) A1
(a)(ii) or \(t=\sqrt{7}.\) A1
Common mistakes
- Mixing up \(ad-bc\) with \(ac-bd\). The determinant multiplies along the leading diagonal first (\(a\times d\)), then subtracts the product of the other diagonal (\(b\times c\)) - keep the entries in the right pairs.
- Forgetting to swap the leading diagonal in the inverse. \(a\) and \(d\) trade places, while \(b\) and \(c\) simply get a minus sign - it's easy to negate the wrong pair under time pressure.
- Trying to invert a matrix with determinant zero. Check \(\det M \neq 0\) before dividing - a zero determinant means the inverse doesn't exist, not that you made an arithmetic slip.
Ready to practise properly?
16 determinant and inverse questions, marked instantly like the real exam.
Quick answers
What is the formula for the determinant of a 2x2 matrix?
For \(M = \begin{pmatrix}a&b\\c&d\end{pmatrix}\), \(\det M = ad - bc\).
What is the formula for the inverse of a 2x2 matrix?
\(M^{-1} = \tfrac{1}{\det M}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\) - swap the leading diagonal, negate the other two entries, then divide by the determinant. It only exists when the determinant is not zero.