How to solve determinate matrix explain with examples

Answer 1

To solve a determinate matrix, you can follow these steps:

1. Start with a square matrix, which means that the number of rows is equal to the number of columns.

2. Calculate the determinate using one of the methods: expansion by cofactors, row reduction, or using the properties of determinants.

3. If using expansion by cofactors, choose a row or column and multiply each element by its corresponding cofactor (the determinate of the matrix formed by removing the row and column of the element). Then, sum these products to get the determinate of the original matrix.

4. Repeat steps 2 and 3 recursively until you obtain a 1x1 matrix, which is just a single number. This number is the determinate of the original matrix.

Here's an example to illustrate this process:

Consider the matrix:
[2 3]
[4 5]

Step 1: Since it is a 2x2 matrix, we can proceed with the calculation.

Step 2: Using expansion by cofactors, we'll choose the first row to calculate the determinate:
D = (2 * det([5])) - (3 * det([4])) = (2 * 5) - (3 * 4) = 10 - 12 = -2

So, the determinate of the matrix [2 3; 4 5] is -2.

Note: The same result would be obtained by using row reduction or applying the properties of determinants, but the approach may differ.