ConvertOnlineMatrix DeterminantCalculator
Matrix A size: x
Answer
How to use this Matrix Determinant Calculator 🤔
Follow these steps to perform Matrix Determinant for the given matrices.
- Enter the matrix size for matrix A.
- Based on the given matrix size, a matrix of input fields appears. Enter the matrix elements.
- As and when you complete entering the matrix elements, Matrix Determinant is calculated, and displayed in the answer section.
Matrix Determinant
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix, such as whether it is invertible. A matrix is invertible if and only if its determinant is non-zero.
The determinant of a 2x2 matrix A, denoted as det(A), is computed as:
det(A) = ad - bc
where A = \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)
Example 1
Let's find the determinant of the following 2x2 matrix:
\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
The determinant of A is:
\( det(A) = (1 \times 4) - (2 \times 3) = 4 - 6 = -2 \)
Example 2
Consider another example with a different 2x2 matrix:
\( A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)
The determinant of A is:
\( det(A) = (5 \times 8) - (6 \times 7) = 40 - 42 = -2 \)
The determinant of a 3x3 matrix A, denoted as det(A), is computed as:
\( det(A) = a~ (ei - fh) - b~ (di - fg) + c~ (dh - eg) \)
where A = \( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \)
Example 3
Let's find the determinant of the following 3x3 matrix:
\( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
The determinant of A is:
\( det(A) = 1(5 \times 9 - 6 \times 8) - 2(4 \times 9 - 6 \times 7) + 3(4 \times 8 - 5 \times 7) \)
\( = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) \)
\( = 1(-3) - 2(-6) + 3(-3) \)
\( = -3 + 12 - 9 \)
\( = 0 \)
These examples demonstrate the basic process of finding the determinant of 2x2 and 3x3 matrices. For larger matrices, the process involves breaking the matrix down into smaller matrices recursively.