Matrices are an essential tool in linear algebra, used for various purposes such as solving systems of linear equations, representing transformations, and analyzing data. One important operation is finding the inverse of a matrix. In this article, we will focus on how to find the inverse of a 2×2 matrix, step by step.

What is a 2×2 matrix?

A 2×2 matrix is a rectangular array of numbers arranged in two rows and two columns. It is denoted as:

[a b]
[c d]

Each element within the matrix can be denoted as a variable, such as a, b, c, and d, representing real numbers.

What is the inverse of a matrix?

The inverse of a matrix is a matrix that, when multiplied with the original matrix, results in the identity matrix. For a 2×2 matrix, the identity matrix is:

[1 0]
[0 1]

Finding the inverse of a matrix is crucial in solving systems of linear equations, as well as various other mathematical applications.

How to find the inverse of a 2×2 matrix?

To find the inverse of a 2×2 matrix, we need to follow these steps:

Step 1: Calculate the determinant
The determinant of a 2×2 matrix is calculated as (ad – bc). In our matrix:

[a b]
[c d]

The determinant is given by (ad) – (bc).

Step 2: Interchange the diagonals
To find the inverse, interchange the diagonals of the matrix, i.e., swap a and d.

[a b]
[c d]

becomes

[d b]
[c a]

Step 3: Change the signs of the remaining elements
Change the signs of the remaining elements, i.e., negate b and c.

[d -b]
[-c a]

Step 4: Multiply by the reciprocal of the determinant
Finally, multiply the entire matrix by the reciprocal of the determinant calculated in Step 1. The reciprocal of determinant (ad – bc) is 1/(ad – bc). Multiply each element of the matrix by this reciprocal value.

[(d/(ad – bc)) (-b/(ad – bc))]
[(-c/(ad – bc)) (a/(ad – bc))]

Example: Finding the inverse of a 2×2 matrix

Let’s consider an example to better understand the steps involved. Suppose we have the matrix:

[3 4]
[2 5]

Step 1: Calculate the determinant: (3*5) – (4*2) = 15 – 8 = 7

Step 2: Interchange the diagonals:

[5 4]
[2 3]

Step 3: Change the signs of the remaining elements:

[5 -4]
[-2 3]

Step 4: Multiply by the reciprocal of the determinant:

[(5/7) (-4/7)]
[(-2/7) (3/7)]

Therefore, the inverse of the given matrix is:

[(5/7) (-4/7)]
[(-2/7) (3/7)]

Calculating the inverse of a 2×2 matrix is a fundamental skill in linear algebra. By following the four steps explained in this article, you can find the inverse of any 2×2 matrix. Remember to calculate the determinant, interchange the diagonals, change the signs of the remaining elements, and multiply by the reciprocal of the determinant. Using this process, you can solve systems of equations, perform matrix operations, and explore various mathematical applications.

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