21. Let \(A\) and \(B\) be upper and lower triangular matrices given by

\(A=\left(\begin{array}{llll} 1& & & & * \\ & 2 & & \\ & & \ddots & \\ 0 & & & &n \end{array}\right)\) and

\(B=\left(\begin{array}{llll} 0 & & & 0 \\ & 1 & & \\ & & \ddots & \\ * & & & n-1 \end{array}\right)\)

Then

Explanation for Question 21:
NA.

22. A homogeneous system of 5 linear equations in 6 variables admits

Explanation for Question 22:
Note that there is a free variable.

23. Let \(A\) be an \(m \times n\) matrix with rank \(m\) and \(B\) be a \(p \times m\) matrix with rank \(p\). What will be the rank of \(B A ?(p < m < n)\)

Explanation for Question 23:
Let LT correspond to \(A\) and \(B\) be \(T\) and \(S\). \(T: \mathbb{R}^n \longrightarrow \mathbb{R}^m\) with \(\rho(T)=m\). \(S: \mathbb{R}^m \longrightarrow \mathbb{R}^p\) with \(\rho(S)=p\). The LT corresponding to \(BA\) is \(\Rightarrow \rho(ToS)=p\). As the composition of two onto maps is onto. This completes the explanation for Question 23.

24. Let \(A\) be an \(n \times n\) matrix and \(b=\left(b_{1}, b_{2}, \cdots, b_{n}\right)^{t}\) be a fixed vector. Consider a system of \(n\)-linear equations \(A x=b\), where \(x=\left(x_{1}, x_{2}, \cdots x_{n}\right)^{\prime}\). Consider the following statements:

A. If rank \(A=n\), the system has a unique solution

B. If rank \(A < n\), the system has infinitely many solutions

C. If \(b=0\), the system has at least one solution

Which of the following is correct?

Explanation for Question 24:
For \(A:\) If \(\rho(A)=n \Rightarrow \rho(A \mid b)=\rho(A)\) \(\Rightarrow A\) has a unique solution. For \(B:\) If \(\rho(A) < n\) and if \(\rho(A \mid b) \neq \rho(A)\) then \(A\) will have no solution. For \(C,\) \(A x=0\) has \(x=0\) as a solution. This completes the explanation for Question 24.

25. Let \(A=\left(a_{i j}\right)\) be an \(n \times n\) matrix such that \(a_{i j}=3\) for \(i\) and \(j\). Then the nullity of \(A\) is

Explanation for Question 25:
\[ \begin{aligned} A=\left(a_{ij}\right) \text { s.t. } a_{ij}=3 \quad \\ \quad \Rightarrow \rho(A)=1 \Rightarrow \eta(A)=n-1 \end{aligned} \] This completes the explanation for Question 25.

26. Let \(A\) be a non-zero matrix of order 8 with \(A^{2}=0\). Then one of the possible values for the rank of \(A\) is

Explanation for Question 26:
If \(A^k=0\) then \(\rho(A) \leq \lfloor \frac{n(k-1)}{k} \rfloor\), where \(A\) is an \(n \times n\) square matrix.

27. The rank of the \(4 \times 4\) skew-symmetric matrix

\(\left[\begin{array}{cccc}0 & 1 & 0 & 1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ -1 & 0 & -1 & 0\end{array}\right]\) is

Explanation for Question 27:
Use row echelon form.

28. Let \(A\) be a \(3 \times 3\) matrix and consider the system of equations

\(A x=\left[\begin{array}{c}1 \\ 0 \\ -1\end{array}\right]\). Then

Explanation for Question 28:
If the system has a unique solution for any \(b\) then the nullspace is trivial. Hence the matrix is invertible.

29. Let \(S\) be the solution space of a set of \(m\) homogeneous linear equations with real coefficients in \(n\) unknowns. If \(A\) is the matrix of this system of equations. Then

Explanation for Question 29:
NA

30. Let \(A\) be an \(n \times n\) matrix over \(\mathbb{R}\). Consider the following statements:

1. Rank \(A=n\)
2. Det \(A \neq 0\)

Then

Explanation for Question 30:
NA