Key takeaway: The determinant is a single number computed from a square matrix; for a 2 × 2 matrix [a b; c d] it equals ad − bc, and a value of zero means no inverse exists.
Grade 12 – PrecalculusN-VM.C.10Expressions and Equations
Definition of Determinant
The determinant boils a square matrix down to one number that reveals key behavior. If it is zero, the matrix squashes the plane onto a line or a point, can be undone by no other matrix, and a system built from it has no unique solution.
Geometrically, the absolute value for a 2 by 2 case is the area of the parallelogram built from its two column arrows; a negative sign signals that the orientation is flipped. For larger matrices it gives a volume. Computing one for 3 by 3 uses expansion along a row, where each entry is multiplied by a smaller determinant with alternating signs.
Formula and notation
Written with bars or as \(\det(A)\):
\[\begin{vmatrix}a & b\\ c & d\end{vmatrix}=ad-bc.\]
For a 3 by 3 matrix expand along the first row: \(a\) times its minor, minus \(b\) times its minor, plus \(c\) times its minor.
Key facts
- A matrix is invertible exactly when this number is not zero.
- Swapping two rows flips its sign.
- Doubling one row doubles the whole value.
- For a product of two matrices, the values multiply.
Determinant example
Find \(\det\begin{bmatrix}5 & -2\\ 3 & 4\end{bmatrix}\) and \(\det\begin{bmatrix}1 & 2 & 0\\ 3 & -1 & 4\\ 2 & 0 & 5\end{bmatrix}\).
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Two by two: \(5\cdot 4-(-2)\cdot 3=20+6=26\).
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Three by three, along row 1: \(1\bigl((-1)(5)-4\cdot 0\bigr)-2\bigl(3\cdot 5-4\cdot 2\bigr)+0\).
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Simplify: \(1(-5)-2(15-8)=-5-14=-19\).
The values are 26 and -19.
Common mistake
Watch out
Sasha computes the first determinant as \(5\cdot 4+(-2)\cdot 3=14\).
Why it is wrong
The formula subtracts the product of the other diagonal: \(ad-bc\). Here \(bc=(-2)(3)=-6\), and subtracting \(-6\) adds 6, giving 26. Keep the minus sign in front of the whole second product.
Related terms and lessons
- Precalculus · Matrices and Matrix OperationsLessonWorksheetQuiz
- Precalculus · Solving Systems with MatricesLessonWorksheetQuiz
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