The determinant of a 2 × 2 matrix
For a 2 × 2 matrix, the determinant compares the products along its two diagonals. Keep the order of the subtraction because it determines the sign
\[\det(A)=ad-bc\]
- Name the entries Use a = 4, b = 2, c = 5 and d = 3 in row order for this example \(\displaystyle A=\begin{bmatrix}4&2\\5&3\end{bmatrix}\)
- Multiply the leading diagonal Multiply the top-left entry by the bottom-right entry \(\displaystyle ad=4\times3=12\)
- Multiply the other diagonal Multiply the top-right entry by the bottom-left entry \(\displaystyle bc=2\times5=10\)
- Subtract in order Subtract the other-diagonal product from the leading-diagonal product \(\displaystyle \det(A)=ad-bc=12-10=2\)
Worked example
Find the determinant of the matrix B.
\[B=\begin{bmatrix}7&-2\\4&3\end{bmatrix}\]
- \(\displaystyle 7\times3=21\)The leading-diagonal product is 7 × 3 = 21.
- \(\displaystyle \left(-2\right)\times4=-8\)The other-diagonal product is (−2) × 4 = −8.
- \(\displaystyle \det(B)=21-\left(-8\right)=29\)Subtracting the negative other-diagonal product gives 21 + 8 = 29.
Common mistakes
Adding the diagonal products
✕\(\displaystyle \det(A)=ad+bc\)✓\(\displaystyle \det(A)=ad-bc\)
The determinant uses a difference between the two diagonal products, not their sumReversing the subtraction
✕\(\displaystyle \det(A)=bc-ad\)✓\(\displaystyle \det(A)=ad-bc\)
The leading-diagonal product comes first. Reversing the subtraction negates the determinant; a zero determinant stays zeroLosing a negative product
✕\(\displaystyle C=\begin{bmatrix}-4&3\\2&5\end{bmatrix},\quad (-4)\times5=20,\quad \det(C)=20-6=14\)✓\(\displaystyle C=\begin{bmatrix}-4&3\\2&5\end{bmatrix},\quad (-4)\times5=-20,\quad \det(C)=-20-6=-26\)
A negative factor makes its product negative, so its sign must stay in the determinant calculationDifficulty levels
introUse positive entries to calculate one determinant from two positive diagonal products
standardKeep signs when negative entries produce a signed determinant
challengeHandle several matrices, including a zero determinant and the inverse condition
examSelect the determinant from options containing addition and reversed-subtraction errors
Questions about this topic
Why is the determinant ad − bc rather than ad + bc?+
The 2 × 2 determinant rule is defined as the leading-diagonal product minus the other-diagonal product. Adding the products gives a different value
What does a determinant of zero tell you?+
A 2 × 2 matrix with determinant zero has no inverse because the inverse rule would require division by zero
How should negative entries be handled?+
Keep the signs while calculating both products. Subtracting a negative product is equivalent to adding its magnitude
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, General Mathematics, chapter 10: Matrices
In this chapter
State the order of a matrix and locate its elementsIdentify row, column, square, symmetric and zero matricesRead information from a labelled data matrixRead, transpose and classify a matrixConstruct a matrix from labelled information or an entry rule10A Matrix notation, types and construction: Solve a matrix equation10A Matrix notation, types and construction: Construct, apply and invert a permutation matrixAdd and subtract matricesFind unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matrices10B Linear operations on matrices: Determine whether an inverse exists and calculate itDecide whether a matrix product is defined and give its orderMultiply matricesEvaluate powers of a square matrixIdentify and construct the identity matrix10C Matrix products, powers and permutations: Construct, apply and invert a permutation matrixFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix10D Determinants, inverses and matrix equations: Determine whether an inverse exists and calculate it10D Determinants, inverses and matrix equations: Solve a matrix equationUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totalsRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixConstruct and analyse relation matrices