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Matrix DeterminantCalculator

Matrix A size: x





Answer




How to use this Matrix Determinant Calculator 🤔

Follow these steps to perform Matrix Determinant for the given matrices.

  1. Enter the matrix size for matrix A.
  2. Based on the given matrix size, a matrix of input fields appears. Enter the matrix elements.
  3. 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.