Question

Suppose Ais \(n \times n\) and the equation \(A{\bf{x}} = {\bf{0}}\) has only the trivial solution. Explain why Ahas npivot columns and Ais row equivalent to \({I_n}\). By Theorem 7, this shows that Amust be invertible. (This exercise and Exercise 24 will be cited in Section 2.3.)

Short answer

Matrix\(A\)is invertible because it has n pivots, and it is row reduced to the identity matrix form.

Step-by-step solution

Write the algorithm for obtaining \({A^{ - 1}}\)

The inverse of an\(m \times m\)matrix A can be computed using theaugmented matrix \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\), where\(I\)is theidentity matrix. Matrix Ahas an inverse only if \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\) is row equivalent to \(\left( {\begin{aligned}{*{20}{c}}I&{{A^{ - 1}}}\end{aligned}} \right)\).

Explain the invertible and linear independence of the matrix

It is given that the matrix equation\(A{\bf{x}} = 0\)has only a trivial solution (at least one solution is 0). It means the columns of an\(n \times n\)matrix Aarelinearly independent, and the matrix has n pivot columns or pivots (no free variable). In this condition, the matrix can be row reduced to the identity matrix.

So, matrix A and identity matrix I are row equivalent, and matrix A isinvertible.