How do you square a 2×2 matrix?

To square a 2×2 matrix, you multiply the matrix by itself. Let’s break this down step-by-step with an example.

Consider a 2×2 matrix A defined as:

A = <span style='font-family: monospace;'>[[a, b],
                [c, d]]</span>

To calculate the square of the matrix, denoted as A², we perform the multiplication:

A² = A * A

This involves applying the formula for matrix multiplication:

A² = <span style='font-family: monospace;'>[[a, b],
                [c, d]] * [[a, b],
                            [c, d]]</span>

To find the entries of A²:

• First entry (Top left): aa + bc
• Second entry (Top right): ab + bd
• Third entry (Bottom left): ca + dc
• Fourth entry (Bottom right): cb + dd

Thus, the squared matrix A² will be:

A² = <span style='font-family: monospace;'>[[aa + bc, ab + bd],
                  [ca + dc, cb + dd]]</span>

In summary, to square a 2×2 matrix, perform matrix multiplication using the standard rules for combining the elements from the rows and columns of the matrices being multiplied.