Determinant of a 3×3 Matrix
Determinant of a 3×3 Matrix is determined by expressing it in terms of 2nd-order determinants. It can be expanded either along rows(R1, R2 or R3) or column(C1, C2 or C3). Consider a matrix A of order 3×3
det(A) = a11(a22a33 – a23a32) – a12(a21a33 – a23a31) + a13(a21a32 – a22a31)
How to Find Determinant of 3×3 Matrix
For calculating the Determinant of 3×3 Matrix use the following step:
Step 1: Multiply the first element a11 of row R1 with (-1)(1 + 1)[(-1)sum of suffixes in a11] and with the second order determinant obtained by deleting the elements of row R1 and C1 of A as a11 lies in R1 and C1. [Tex](-1)^{1 + 1}a_{1}\begin{vmatrix} b_{2} & b_{3} \\ c_{2} & c_{3} \end{vmatrix}[/Tex]
Step 2: Similarly, multiply the second element of the first rowR1, with the determinant obtained after deleting the first row and second column. [Tex](-1)^{1 + 2}a_{12}\begin{vmatrix} b_{1} & b_{3} \\ c_{1} & c_{3} \end{vmatrix}[/Tex]
Step 3: Multiply the third element of row R1 with the determinant obtained after deleting the first row and third column. [Tex](-1)^{1 + 3}a_{3}\begin{vmatrix} b_{1} & b_{2} \\ c_{1} & c_{2} \end{vmatrix}[/Tex]
Step 4: Now the expansion of the determinant of A, that is |A| can be written as |A| = [Tex](-1)^{1 + 1}a_{1}\begin{vmatrix} b_{2} & b_{3} \\ c_{2} & c_{3} \end{vmatrix} + (-1)^{1 + 2}a_{12}\begin{vmatrix} b_{1} & b_{3} \\ c_{1} & c_{3} \end{vmatrix} + (-1)^{1 + 3}a_{3}\begin{vmatrix} b_{1} & b_{2} \\ c_{1} & c_{2} \end{vmatrix}[/Tex]
Similarly, in this way, we can expand it along any row and any column.
Example: Evaluate the determinant det(A) = [Tex]\begin{vmatrix} 1 & 3 & 0 \\ 4 & 1 & 0 \\ 2 & 0 & 1 \end{vmatrix}[/Tex]
Solution:
We see that the third column has most number of zeros, so it will be easier to expand along that column.
det(A) = [Tex](-1)^{1 + 3}0\begin{vmatrix}4 & 1 \\ 2 & 0 \end{vmatrix} + (-1)^{2 + 3}0\begin{vmatrix}1 & 3 \\ 2 & 0 \end{vmatrix} + (-1)^{1 + 3}1\begin{vmatrix}1 & 3 \\ 4 & 1 \end{vmatrix} \\ = -11[/Tex]
Determinant of a Matrix with Solved Examples
Determinant of a Matrix is defined as the function that gives the unique output (real number) for every input value of the matrix. Determinant of the matrix is considered the scaling factor that is used for the transformation of a matrix. It is useful for finding the solution of a system of linear equations, the inverse of the square matrix, and others. The determinant of only square matrices exists.
Table of Content
- Determinant of Matrix Calculator
- Definition of Determinant of Matrix
- Determinant of a 1×1 Matrix
- Determinant of 2×2 Matrix
- Determinant of a 3×3 Matrix
- Determinant of 4×4 Matrix
- Determinant of Identity Matrix
- Determinant of Symmetric Matrix
- Determinant of Skew-Symmetric Matrix
- Determinant of Inverse Matrix
- Determinant of Orthogonal Matrix
- Physical Significance of Determinant
- Laplace Formula for Determinant
- Properties of Determinants of Matrix