Matrices & Determinants
Matrix inverse and determinant condition
nta_pyq_2025_apr
Grade 12
Question:
Let A = [$\alpha$-1 ],$\alpha$> 0 , such that$det(A) = 0$and$\alpha$+$\beta$= 1. If I denotes 2 $\times$ 2 identity matrix, then the 6$\beta$matrix$(1 + A)$is: 8$4 -1$
[ ]$6 -1$$257 -64$
[ ]$514 -127$$1025 -511$
[ ]$2024 -1024$$766 -255$
[ ]$1530 -509$
Step-by-Step Solution
Key Concept: Apply the matrix property for matrix inverse and determinant condition and reduce it to determinant or parameter equations.
|A| = 0 (4)$\alpha$$\beta$+$6 = 0$$\alpha$$\beta$= -6$\alpha$+$\beta$= 1 $\Rightarrow$$\alpha$= 3,$\beta$= -2$3 -1$A = [ ]$6 -2$2$3 -1$$3 -1$$3 -1$A = [ ][ ] = [ ]$6 -2$$6 -2$$6 -2$2 ∴$A = A$2 3 4 5$A = A = A = A = A$8$(I + A)$8 7 8 6 8$8 = I$+$C_{1}$A +$C_{2}$A +$\ldots$. . +$C_{8}$A 8 8$8 = I + A$($C_{1}$+$C_{2}$+$\ldots$. . +$C_{8}$)$8 = I + A$$(2 - 1)$1 0$765 -255$= [ ] + [ ] 0 1$1530 -510$$766 -255$= [ ]$1530 -509$
Correct Answer: 4