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 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind 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 matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals