Matrix Solver, Determinant 2 by 2 Matrix

Finds 2 by 2 matrix determinant in this form \[ \begin{pmatrix} a \ \ \ b \\ c \ \ \ d \end{pmatrix} \]

2 by 2 matrix determinants

2 by 2 matrix determinants calculator

The MathCrave 2 by 2 Matrix determinants calculator computes the determinant of a 2x2 matrix by multiplying the numbers on the main diagonal and subtracting the product of the numbers on the other diagonal.

How To Use 2 by 2 Matrix Determinants calculator

matrix A Element

Enter all the element of A like so a11, a12, a21, a22

Matrix 2 by 2 determinant result

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About MathCrave 2 by 2 Matrix Determinants?

MathCrave 2 by 2 Matrix determinants calculates the determinant of a 2x2 matrix by multiplying the values in the main diagonal (top-left to bottom-right) and subtracting the product of the other diagonal (top-right to bottom-left).

Understanding the Formula for Matrix Determinant 2 by 2

The determinant of a 2 by 2 matrix [A] is given by the formula:

  • |A| = a*d - b*c

where [A] = [[a, b], [c, d]] represents the given 2 by 2 matrix.

Detailed Guide to Solve 2 by 2 Matrix Determinant

To solve for the determinant of a 2 by 2 matrix, follow these steps:

  • Step 1: Identify the matrix [A] = [[a, b], [c, d]].

  • Step 2: Apply the determinant formula: |A| = a*d - b*c.

  • Step 3: Substitute the respective values of a, b, c, and d into the formula.

  • Step 4: Perform the necessary arithmetic operations.

  • Step 5: The result obtained is the determinant of the given 2 by 2 matrix.

2 by 2 Matrix Determinant Worked Example

Example: Calculate the determinant of the matrix [B] = [[3, 2], [7, 5]].

Solution:

  • Step 1: Identify the matrix

  • [B] = [[3, 2], [7, 5]].

  • Step 2: Apply the determinant formula:

  • |B| = a*d - b*c.

  • Step 3: Substitute the values of a, b, c, and d:

  • |B| = (3*5) - (2*7).

  • Step 4: Perform the arithmetic operations:

  • |B| = 15 - 14.

  • Step 5: The determinant of matrix B is

  • |B| = 1.

  • Therefore, the determinant of the matrix

  • [B] = [[3, 2], [7, 5]] is 1.

Theorem Binding Matrix Determinants

The 2 by 2 matrix determinant follows the theorem known as the "Laplace's theorem" or "Laplace expansion theorem." It states that the determinant of a matrix can be calculated by expanding along any row or column and taking the sum of the products of the elements with their corresponding cofactors.

However, for 2 by 2 matrices, the determinant formula suffices, where the product of the diagonal elements is subtracted from the product of the off-diagonal elements.

Practice Questions on Matrix Determinants 2 by 2

  • Find the determinant of the matrix A = [3 2; -1 4].

  • Determine the value of the determinant for the matrix B = [5 -2; 7 -1].

  • Calculate the determinant for the matrix C = [-2 0; 6 3].

  • Find the determinant of the matrix D = [1/2 3; -5/4 2].

  • Determine the value of the determinant for the matrix E = [0 -1/2; 4 1/3].

  • Calculate the determinant for the matrix F = [2 -5; -3 7/4].

  • Find the determinant of the matrix G = [6 4; -2 -3].

  • Determine the value of the determinant for the matrix H = [1 2; 3/2 4/3].

  • Calculate the determinant for the matrix I = [-7 1; 1/5 -3/2].

  • Find the determinant of the matrix J = [0 3; -2 0].

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