❖ In the lesson, we learn about the negative powers of a matrix.

❖ The tutorial starts by calculating \(A^{-1}, A^{-2},\) and \(A^{-3},\) then moves on to formulating a general formula for \(A^n\) where \(n\) is any negative integer. By using this formula, we compute \(A^{-2025}\)

❖ The negative power \(A^n\) of a matrix \(A\), where \(n\) is a negative integer is defined as the matrix product of \(n\) copies of the inverse of \(A\)

To find the negative power of \(A\),

the inverse of \(A\) has to be exist.

(so, \(A^{-1}\) has to be exist).

To find \(A^{-1}\),

A must be a square matrix.

This means that

the number of rows \(=\) the number of columns

(for example, its dimensions could be \(2 \times 2\), \(3 \times 3\), etc.).

\(A^0\) is defined to be

\(A^0=I_n\)

where \(I_n\) is an identity matrix with the same size as \(A\).

❖ The number of rows and columns that a matrix has is called its size, its order, or its dimension.