Question

Let \(A = [\begin{array}{cc}-5 & -3 \\ 2 & 1\end{array}]\). The Row transformation \(R_1\rightarrow R_1+3R_2\) will transform matrix A into

• A. an upper triangular matrix
• B. a lower triangular matrix
• C. an identity matrix
• D. a singular matrix

Hint:

Carry out the row operation and read off the shape of the new matrix.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In the operation \(R_1 \to R_1 + 3R_2\) we replace the first row by the first row plus three times the second row. The second row stays the same.

Step 2: Detailed Explanation:
\(R_1 = (-5, -3)\) and \(3R_2 = (6, 3)\).
New \(R_1 = (-5 + 6, -3 + 3) = (1, 0)\).
\[ A \to \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \]

Step 3: Classify.
The entry above the diagonal is 0, so this is a lower triangular matrix. It is not upper triangular (the entry 2 below the diagonal is non-zero), not an identity (because of the 2), and its determinant is \(1\), so it is not singular.

Final Answer:
The matrix becomes lower triangular, option (B). \[ \boxed{\text{Lower triangular matrix}} \]