Elementary Transformations

Q.

$A=\left[\begin{array}{ccc}1& 5& 2\\ 3& 1& 4\\ 2& 2& 3\end{array}\right]$ .The following elementary transformations are applied on the matrix A in the given order. What will be the resultant matrix?

1. $R_2$ $\to$ $R_2$ – $R_3$
2. $R_1$ $\to$ $R_1–R_2$
3. $R_3$ $\to$ $R_3–2R_2$

1. $\left[\begin{array}{ccc}0& 6& 1\\ 2& -1& 2\\ 0& -4& 1\end{array}\right]$

2. $\left[\begin{array}{ccc}0& 6& 2\\ 2& 1& 1\\ 0& -4& 1\end{array}\right]$

3. $\left[\begin{array}{ccc}0& 6& 2\\ 1& 1& 1\\ 0& -4& 1\end{array}\right]$

4. $\left[\begin{array}{ccc}0& 6& 1\\ 1& -1& 1\\ 0& 4& 1\end{array}\right]$

Q.

A matrix resulted from elementary row and column transformation is not equivalent to the original matrix as it happens for just row or just column transformations. Given Equivalent means the matrices can be transformed into one another by a combination of elementary row and column operations.

1. True

2. False

Q.

Which of the following can be treated as an Elementary Row transformation.

1. Row 1 → Row 1 + Row 3

2. Row 2 → Row 2 + 2 Row 1

3. Row 2 → Row 2 + 2 Row 2

4. Row 3 → Row 3 - Row 3

Q.

The following matrix can be converted to a Identity matrix using elementary row transformations $\left[\begin{array}{ccc}1& 2& 2\\ 2& 5& 5\\ -1& 1& 1\end{array}\right]$

1. True

2. False

Q.

Multiplying every element of a row of a given matrix by an integer is an elementary transformation.

1. True

2. False

Q. Suppose $A$ is any $3\times 3$ non-singular matrix and $(A-3I)(A-5I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+\beta A^{-1}=4I$, then $\alpha +\beta$ is equal to :

1. $13$
2. $7$
3. $12$
4. $8$

Q. covert it into an upper triangular matrix
Convert $\left[\begin{array}{c}1\\ 2\end{array}\begin{array}{c}-1\\ 3\end{array}\right]$ into an identity matrix by suitable row transformations.

Q. $A=\left[\begin{array}{ccc}2& 2& 1\\ 0& 1& 4\\ 0& 2& 6\end{array}\right]$, $B=\left[\begin{array}{ccc}2& 2& 1\\ 0& 1& 4\\ 0& 0& 1\end{array}\right]$

1. $R_3\to R_3-3R_1$, $R_3\to R_2-R_1$
2. $R_3\to R_1-2R_2$, $R_3\to (R_3\times -2)$
3. $R_2\to R_2-2R_2$, $R_3\to (R_3÷2)$
4. $R_3\to R_3-2R_2$, $R_3\to (R_3÷-2)$

Q. Apply the given elementary transformation of the following matrix.
Transform $\left[\begin{array}{ccc}1& -1& 2\\ 2& 1& 3\\ 3& 2& 4\end{array}\right]$ into an upper triangular matrix by suitable column transformations.

Q. Convert $\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ into on identity matrix by suitable raw transformation.