(a+b)2=a2+b2+2ab Matrix

4 min read Jun 16, 2024

The Square of a Matrix Sum: (A + B)² = A² + B² + 2AB

In linear algebra, understanding the behavior of matrices under various operations is crucial. One such operation is squaring a sum of matrices, represented by the equation:

(A + B)² = A² + B² + 2AB

This equation holds true only for certain types of matrices and under specific conditions. It is important to note that this is not a universal law for all matrices.

Understanding the Equation

Let's break down the equation and the conditions for it to be valid:

A and B: These represent square matrices of the same order (i.e., they have the same number of rows and columns).
A² and B²: These represent the square of each matrix. Squaring a matrix means multiplying it by itself.
AB: This represents the matrix product of A and B.
2AB: This represents the scalar multiplication of 2 with the product of matrices A and B.

Conditions for the Equation to Hold True

The equation (A + B)² = A² + B² + 2AB is not universally true for all matrices. It holds true under the following conditions:

Commutativity: The matrices A and B must commute, meaning AB = BA. This condition is essential because matrix multiplication is generally not commutative.
Square Matrices: Both A and B must be square matrices of the same order.

Examples

Example 1: Let's consider two matrices:

A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]

Verification:

AB = [[19, 22], [43, 50]]
BA = [[19, 22], [43, 50]]

Since AB = BA, the matrices A and B commute. Therefore, the equation (A + B)² = A² + B² + 2AB holds true in this case.

Example 2: Let's consider two matrices:

A = [[1, 2], [3, 4]]
B = [[5, 0], [0, 6]]

Verification:

AB = [[5, 12], [15, 24]]
BA = [[5, 0], [15, 24]]

Since AB ≠ BA, the matrices A and B do not commute. Therefore, the equation (A + B)² = A² + B² + 2AB does not hold true in this case.

Conclusion

The equation (A + B)² = A² + B² + 2AB holds true only when the matrices A and B commute and are both square matrices of the same order. It's crucial to remember this condition before applying the equation. Failing to do so can lead to incorrect results.