# Linear Algebra Questions and Answers – Rank of Matrix in PAQ and Normal Form

This set of Engineering Mathematics Assessment Questions and Answers focuses on “Rank of Matrix in PAQ and Normal Form”.

1. Reduce the given matrix to normal form, and hence find its rank.
$$\begin{bmatrix}1& 2& -2& 3\\ 2& 5& -4& 6\\ -1& -3& 2& -2\\ 2& 4& -1& 6\end{bmatrix}$$
a) 1
b) 4
c) 3
d) 2

Explanation: In this Question we have,
A=$$\begin{bmatrix}1& 2& -2& 3\\ 2& 5& -4& 6\\ -1& -3& 2& -2\\ 2& 4& -1& 6\end{bmatrix}$$
By R2-2R1 and R3+R1 and R4-2R1
A=$$\begin{bmatrix}1& 2& -2& 3\\ 0& 1& 0 & 0\\ 0& -1& 0 & 1\\ 0& 0 & 3& 0\end{bmatrix}$$
By C2-2C1, C3+2C1, C4-3C1
A=$$\begin{bmatrix}1& 0& 0 & 0\\ 0& 1 & 0 & 0\\ 0& -1 & 0 &1 \\ 0& 0 & 3 &0 \end{bmatrix}$$
By R2+R3
A=$$\begin{bmatrix}1& 0 & 0 &0 \\0 & 1 & 0 & 0\\0 & 0 & 0& 1\\ 0& 0 & 3 & 0\end{bmatrix}$$
By C34
A=$$\begin{bmatrix}1& 0 & 0 &0 \\0 & 1 & 0 &0 \\ 0& 0 & 1 & 0\\ 0& 0 & 0 & 3\end{bmatrix}$$
By $$\frac{1}{3}$$ C4
A=$$\begin{bmatrix}1& 0 & 0 &0 \\0 & 1 & 0 &0 \\ 0& 0 & 1 & 0\\ 0& 0 & 0 & 1\end{bmatrix}$$
This is in Normal Form.
Hence the rank of the given Matrix is 4.

2. Find the value of p for which, the rank of the given matrix is 1.
$$\begin{bmatrix}3&p&p\\p&3&p\\p&p&3\end{bmatrix}$$
a) 4
b) 2
c) 3
d) 1

Explanation: In this Question we have,
A=$$\begin{bmatrix}3&p&p\\p&3&p\\p&p&3\end{bmatrix}$$
If we put the value of p=3,
The matrix becomes:
A=$$\begin{bmatrix}3&3&3\\3&3&3\\3&3&3\end{bmatrix}$$
By R2-R1 and R3-R1
A=$$\begin{bmatrix}3&3&3\\0&0&0\\0&0&0\end{bmatrix}$$
Thus, the rank of the given matrix is 1.
Therefore, the value of p must be 3.

3. Find the value of non singular matrices P and Q, such that PAQ is in the normal form, where A is
$$\begin{bmatrix}1&1&2\\1&2&3\\0&-1&-1\end{bmatrix}$$

a) $$\begin{bmatrix}1&0&0\\-1&1&0\\-1&1&1\end{bmatrix}$$, $$\begin{bmatrix}1&-1&-1\\0&1&-1\\0&0&1\end{bmatrix}$$
b)$$\begin{bmatrix}1&1&0\\-1&1&0\\-1&1&1\end{bmatrix}$$, $$\begin{bmatrix}1&-1&-1\\0&1&-1\\0&0&1\end{bmatrix}$$
c)$$\begin{bmatrix}1&0&0\\-1&1&0\\-1&1&1\end{bmatrix}$$, $$\begin{bmatrix}1&-1&1\\0&1&-1\\0&0&1\end{bmatrix}$$
d)$$\begin{bmatrix}1&0&0\\-1&1&0\\-1&0&1\end{bmatrix}$$, $$\begin{bmatrix}1&-1&-1\\0&1&-1\\0&0&-1\end{bmatrix}$$

Explanation: In this Question we have,
A=$$\begin{bmatrix}1&1&2\\1&2&3\\0&-1&-1\end{bmatrix}$$

Let A=I3AI3

$$\begin{bmatrix}1&1&2\\1&2&3\\0&-1&-1\end{bmatrix}$$=$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$A$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$

By C2-C1 and C3-2C1

$$\begin{bmatrix}1&0&0\\1&1&1\\0&-1&-1\end{bmatrix}$$=$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$A$$\begin{bmatrix}1&-1&-2\\0&1&0\\0&0&1\end{bmatrix}$$

By R2-R1
$$\begin{bmatrix}1&0&0\\0&1&1\\0&-1&-1\end{bmatrix}$$=$$\begin{bmatrix}1&0&0\\-1&1&0\\0&0&1\end{bmatrix}$$A$$\begin{bmatrix}1&-1&-2\\0&1&0\\0&0&1\end{bmatrix}$$

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By C3-C2

$$\begin{bmatrix}1&0&0\\0&1&0\\0&-1&0\end{bmatrix}$$=$$\begin{bmatrix}1&0&0\\-1&1&0\\0&0&1\end{bmatrix}$$A$$\begin{bmatrix}1&-1&-1\\0&1&-1\\0&0&1\end{bmatrix}$$

By R3+R2

$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}$$=$$\begin{bmatrix}1&0&0\\-1&1&0\\-1&1&1\end{bmatrix}$$A$$\begin{bmatrix}1&-1&-1\\0&1&-1\\0&0&1\end{bmatrix}$$

This is in Normal Form.

$$\begin{bmatrix}I2&0\\0&0\end{bmatrix}$$=PAQ

P=$$\begin{bmatrix}1&0&0\\-1&1&0\\-1&1&1\end{bmatrix}$$ and
Q=$$\begin{bmatrix}1&-1&-1\\0&1&-1\\0&0&1\end{bmatrix}$$.

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