Question

(a) Consider an matrix A such that ${\mathit{A}}^{\mathbf{\Gamma }}\mathit{A}\mathbf{=}{\mathit{I}}_{\mathbf{m}}$. It is necessarily true that? Explain.

(b) Consider an $\mathit{n}\mathbf{\times }\mathit{n}$matrix A such that ${\mathit{A}}^{\mathbf{T}}\mathit{A}\mathbf{=}{\mathit{I}}_{\mathbf{n}}$. Is it necessarily true that $\mathit{A}{\mathit{A}}^{\mathbf{T}}\mathbf{=}{\mathit{I}}_{\mathbf{n}}$? Explain.

Short answer

1. Not always true for condition (a).
2. Always true for condition (b).

Step-by-step solution

Determine AAT≠I3.

To obtain $A{A}^T\ne I_3$consider the matrix below.

$A=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]$

In the matrix A. performing product,

$A{A}^T=I_2$.

Which shows that $A{A}^T\ne I_3$

Thus, according to condition (a) it is not necessarily true that $A{A}^T=I_3$.

Consider the definition of inverse matrix.

The inverse of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^-1.

And the formula of inverse matrix is given below.

$A^{-1}=\frac{1}{\left|A\right|}.AdjA$

For example,

$$\begin{gathered} A=\left[\begin{array}{cc}1& -2\\ 3& 4\end{array}\right] \\ A=\left[\begin{array}{cc}0.4& 0.2\\ -0.3& 0.1\end{array}\right] \end{gathered}$$

Thus, by consider the definition of inverse matrix above the condition (b) is always true.

Hence, condition (a) is not necessarily true and condition (b) is always true.