Question

If \[ \begin{bmatrix} 2x+1 & 5x \\ 0 & y^2+1 \end{bmatrix} = \begin{bmatrix} x+3 & 10 \\ 0 & 26 \end{bmatrix} \] then the possible values of \(x+y\) are:

• A. \(2\ \text{and}\ 5\)
• B. \(5\ \text{and}\ -1\)
• C. \(7\ \text{and}\ -3\)
• D. \(2\ \text{and}\ -5\)

Hint:

For equal matrices: \[ \text{Corresponding elements must be equal.} \]

Answer 1

The Correct Option is C

Two matrices are equal if their corresponding elements are equal.
Step 1: Compare corresponding elements From: \[ 2x+1=x+3 \] we get: \[ x=2 \] Also: \[ 5x=10 \] Substituting: \[ x=2 \] gives: \[ 10=10 \] which is correct.
Step 2: Find value of \(y\) From: \[ y^2+1=26 \] \[ y^2=25 \] \[ y=\pm5 \] Thus: \[ y=5 \quad \text{or} \quad y=-5 \]
Step 3: Find possible values of \(x+y\) When: \[ x=2,\ y=5 \] \[ x+y=7 \] When: \[ x=2,\ y=-5 \] \[ x+y=-3 \] Therefore, possible values are: \[ 7\ \text{and}\ -3 \] Option analysis:
• Option (A): Incorrect
• Option (B): Incorrect
• Option (C): Correct
• Option (D): Incorrect Hence: \[ \boxed{\text{(C)}} \]