Finds 2 by 2 matrix determinant in this form \[ \begin{pmatrix} a \ \ \ b \\ c \ \ \ d \end{pmatrix} \]
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).
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.
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.
Example: Calculate the determinant of the matrix [B] = [[3, 2], [7, 5]].
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.
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.
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