Matrices & Determinants
Polynomial equation in a matrix
nta_pyq_2025_apr
Grade 12
Question:
Let A be a 3 $\times$ 3 real matrix such that A$(A - 2I$$)- 4($$A - I) = O$, where I and O are the identity and null 2 matrices, respectively. If A =$\alphaA$+$\betaA$+$\gammaI$, where$\alpha$,$\beta$and$\gamma$are real constants, then$\alpha$+$\beta$+$\gamma$is equal to: 5 2
Step-by-Step Solution
Key Concept: Apply the matrix property for polynomial equation$i_n$a matrix and reduce it to determinant or parameter equations.
3 2$A - 2A - 4A + 4I = 0$3 2 (1)$A = 2A + 4A - 4I$4 3 2$A = 2A + 4A - 4A$2$2 = 2$$(2A + 4A - 4I$$) + 4A - 4A$4 2$A = 8A + 4A - 8I$5 3 2$A = 8A + 4A - 8A$2$2 = 8$$(2A + 4A - 4I$$) + 4A - 8A$5 2$A = 20A + 24A - 32I$∴$\alpha$= 20,$\beta$= 24,$\gamma$= -32 ∴$\alpha$+$\beta$+$\gamma$= 12
Correct Answer: 1