Matrices & DeterminantsmediumFree

Matrices & Determinants: Matrix Given Pmatrix Pmatrix Determinant

JEE Maths question with a full step-by-step solution.

Question
Matrix AA is given by A=(31128)A = \begin{pmatrix} 3 & 11 \\ 2 & 8 \end{pmatrix}. Then the determinant of A2011−5A2010A^{2011} - 5A^{2010} is:
A−7⋅22012-7\cdot 2^{2012}correct
B220122^{2012}
C−7⋅22010-7\cdot 2^{2010}
D7⋅220107\cdot 2^{2010}
Solution
Step 1: Compute det⁡A\det A
det⁡A=(3)(8)−(11)(2)=2\det A = (3)(8) - (11)(2) = 2
 Step 2: Factor the expression
A2011−5A2010=A2010(A−5I)A^{2011} - 5A^{2010} = A^{2010}(A - 5I)
det⁡(A2011−5A2010)=(det⁡A)2010 det⁡(A−5I)=22010 det⁡(A−5I)\det(A^{2011} - 5A^{2010}) = (\det A)^{2010}\,\det(A - 5I) = 2^{2010}\,\det(A - 5I)
 Step 3: Evaluate det⁡(A−5I)\det(A - 5I)
A−5I=(−21123),det⁡(A−5I)=(−2)(3)−(11)(2)=−28A - 5I = \begin{pmatrix} -2 & 11 \\ 2 & 3 \end{pmatrix}, \quad \det(A - 5I) = (-2)(3) - (11)(2) = -28
 Step 4: Combine
det⁡(A2011−5A2010)=22010(−28)=−7⋅22012\det(A^{2011} - 5A^{2010}) = 2^{2010}(-28) = -7\cdot 2^{2012}
 Answer: (1)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Matrices & Determinants · easy
If AA is a 3×33 \times 3 matrix and det⁡A=5\det A = 5, then det⁡(Adj(Adj A))\det\big(\text{Adj}(\text{Adj}\,A)\big) equals:
Matrices & Determinants · easy
The equation (12213434k)(xyz)=(000)\begin{pmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} has a solution (x,y,z)(x, y, z) besides (0,0,0)(0, 0, 0). The value of kk equals:
Matrices & Determinants · easy
AA is an involutory matrix given by A=(01−14−343−34)A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}, then the inverse of A2\dfrac{A}{2} will be:
Matrices & Determinants · easy
A square matrix PP satisfies P2=I−PP^2 = I - P, where II is the identity matrix. If Pn=5I−8PP^n = 5I - 8P, then nn is:

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath