- Calculate determinants of 2×2 and 3×3 matrice
Evaluate the Determinant of a 2×2 Matrix
A determinant is a real number calculated from the entries of a square matrix. Determinants have practical applications in calculating area and volume, solving systems of equations, and even cryptography—where encoded messages can only be decrypted using an invertible matrix and its determinant.
For us, the determinant serves a specific purpose: it tells us whether a matrix is invertible. If the determinant equals zero, the matrix is not invertible. If it’s nonzero, we can find an inverse.
the determinant of a 2 × 2 matrix
The determinant of a [latex]2\text{ }\times \text{ }2[/latex] matrix, given
is defined as
You’ll see determinants written in several ways:
- [latex]\text{det}(A)[/latex]
- [latex]|A|[/latex] (vertical bars replacing the matrix brackets)
These all mean the same thing—the determinant of matrix [latex]A[/latex].
[latex]A=\left[\begin{array}{cc}5& 2\\ -6& 3\end{array}\right][/latex]
Evaluate the Determinant of a 3 × 3 Matrix
Finding the determinant of a [latex]2 \times 2[/latex] matrix is straightforward, but the [latex]3 \times 3[/latex] case requires more steps. We’ll use a method called diagonal expansion.
The process: augment the matrix by repeating its first two columns on the right, creating a [latex]3 \times 5[/latex] array. Then calculate products along diagonals—adding products that go down-right and subtracting products that go up-right.
Given matrix [latex]A=\left[\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}\right][/latex]
Step 1: Augment [latex]A[/latex] with its first two columns:
[latex]\text{det}(A) = \left|\begin{array}{ccc}a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\end{array}\right| \left.\begin{array}{cc}a_1 & b_1\\ a_2 & b_2\\ a_3 & b_3\end{array}\right|[/latex]
Step 2: Add products down the three diagonals (upper left to lower right):
- First diagonal: [latex]a_1 b_2 c_3[/latex]
- Second diagonal: [latex]b_1 c_2 a_3[/latex]
- Third diagonal: [latex]c_1 a_2 b_3[/latex]
Step 3: Subtract products up the three diagonals (lower left to upper right):
- First diagonal: [latex]a_3 b_2 c_1[/latex]
- Second diagonal: [latex]b_3 c_2 a_1[/latex]
- Third diagonal: [latex]c_3 a_2 b_1[/latex]
Result:
[latex]|A| = a_1 b_2 c_3 + b_1 c_2 a_3 + c_1 a_2 b_3 - a_3 b_2 c_1 - b_3 c_2 a_1 - c_3 a_2 b_1[/latex]
Think of it as:
- Add the three products going down-right (↘)
- Subtract the three products going up-right (↗)
This pattern makes the calculation systematic and reduces errors.
[latex]A=\left[\begin{array}{ccc}0& 2& 1\\ 3& -1& 1\\ 4& 0& 1\end{array}\right][/latex]
[latex]\mathrm{det}\left(A\right)=\left\rvert\begin{array}{ccc}1& -3& 7\\ 1& 1& 1\\ 1& -2& 3\end{array}\right\rvert[/latex]