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Matrix Determinant calculator lets you calculate the determinant of a given matrix, with step by step calculations. We shall transform the matrix into an upper triangular matrix, and then the determinant of the matrix is equal to the product of diagonal elements.
Examples to use Matrix Determinant Calculator
1. Find the determinant of the following 3×3 matrix.
Matrix \({\ A = \begin{bmatrix}1 & 2 & 2\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} }\)
Solution
Since the given matrix is of size 3×3, follow these steps with the Matrix Determinant Calculator.
- Enter value of 3 in input field: m or n.
- A matrix with specified size of 3×3 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}1 & 2 & 2\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} \)
Transform A to Upper Triangular Matrix
Applying \( R2 → R2-(4)R1\)
\( = \begin{bmatrix}1 & 2 & 2\\0 & -3 & -2\\7 & 8 & 9\end{bmatrix} \)
Applying \( R3 → R3-(7)R1\)
\( = \begin{bmatrix}1 & 2 & 2\\0 & -3 & -2\\0 & -6 & -5\end{bmatrix} \)
Applying \( R3 → R3-(2)R2\)
\( = \begin{bmatrix}1 & 2 & 2\\0 & -3 & -2\\0 & 0 & -1\end{bmatrix} \)
The given Matrix A has transformed into an upper triangular matrix.
Calculating Determinant of Matrix A
The determinant of a triangular matrix is the product of the diagonal elements.
\(|A| = 1 × (-3) × (-1) = 3 \)
Result
Therefore, the determinant of given matrix A is
\(|A| = 3\)
2. Find the determinant of the following 4×4 matrix.
Matrix \({\ A = \begin{bmatrix}0 & 2 & 3 & 4\\5 & 2 & 1 & 2\\0 & 2 & 3 & 0\\2 & 3 & 0 & 0\end{bmatrix} }\)
Solution
Since the given matrix is of size 4×4, follow these steps with the Matrix Determinant Calculator.
- Enter value of 4 in input field: m or n.
- A matrix with specified size of 4×4 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}0 & 2 & 3 & 4\\5 & 2 & 1 & 2\\0 & 2 & 3 & 0\\2 & 3 & 0 & 0\end{bmatrix} \)
Transform A to Upper Triangular Matrix
Pivot element in Row 1 is 0. Therefore, swapping rows R1 and R2.
Applying \(R1 ↔ R2\)
\( = \begin{bmatrix}5 & 2 & 1 & 2\\0 & 2 & 3 & 4\\0 & 2 & 3 & 0\\2 & 3 & 0 & 0\end{bmatrix} \)
Applying \( R4 → R4-(\frac{2}{5})R1\)
\( = \begin{bmatrix}5 & 2 & 1 & 2\\0 & 2 & 3 & 4\\0 & 2 & 3 & 0\\0 & \frac{11}{5} & \frac{-2}{5} & \frac{-4}{5}\end{bmatrix} \)
Applying \( R3 → R3-(1)R2\)
\( = \begin{bmatrix}5 & 2 & 1 & 2\\0 & 2 & 3 & 4\\0 & 0 & 0 & -4\\0 & \frac{11}{5} & \frac{-2}{5} & \frac{-4}{5}\end{bmatrix} \)
Applying \( R4 → R4-(\frac{11}{10})R2\)
\( = \begin{bmatrix}5 & 2 & 1 & 2\\0 & 2 & 3 & 4\\0 & 0 & 0 & -4\\0 & 0 & \frac{-37}{10} & \frac{-26}{5}\end{bmatrix} \)
Pivot element in Row 3 is 0. Therefore, swapping rows R3 and R4.
Applying \(R3 ↔ R4\)
\( = \begin{bmatrix}5 & 2 & 1 & 2\\0 & 2 & 3 & 4\\0 & 0 & \frac{-37}{10} & \frac{-26}{5}\\0 & 0 & 0 & -4\end{bmatrix} \)
The given Matrix A has transformed into an upper triangular matrix.
Calculating Determinant of Matrix A
The determinant of a triangular matrix is the product of the diagonal elements.
\(|A| = 5 × 2 × \frac{-37}{10} × (-4) = 148 \)
Result
Therefore, the determinant of given matrix A is
\(|A| = 148\)