Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Matrices and Determinants question

2018 · 15 Apr · Shift 1 · Q29
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Matrices and Determinants
  5. /2018 · 15 Apr · Shift 1 · Q29

Matrices and Determinants question

2018 · 15 Apr · Shift 1 · Q29

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
Let AAA be a matrix such that A.[1203]A.\left[ {\begin{matrix} 1 & 2 \\ 0 & 3 \\ \end{matrix} } \right]A.[10​23​] is a scalar matrix and |3A| = 108. Then A2 equals :
  1. A
    [4−32036]\left[ {\begin{matrix} 4 & { - 32} \\ 0 & {36} \\ \end{matrix} } \right][40​−3236​]
  2. B
    [360−324]\left[ {\begin{matrix} {36} & 0 \\ { - 32} & 4 \\ \end{matrix} } \right][36−32​04​]
  3. C
    [40−3236]\left[ {\begin{matrix} 4 & 0 \\ { - 32} & {36} \\ \end{matrix} } \right][4−32​036​]
  4. D
    [36−3204]\left[ {\begin{matrix} {36} & { - 32} \\ 0 & 4 \\ \end{matrix} } \right][360​−324​]
View written solutionFree

Correct answer: D

  1. Let B=[1203]B=\begin{bmatrix}1&2\\0&3\end{bmatrix}B=[10​23​] and suppose ABABAB is a scalar matrix.

A scalar matrix has the form λI\lambda IλI, so AB=λI.AB=\lambda I.AB=λI. Hence, A=λB−1.A=\lambda B^{-1}.A=λB−1.

  1. Compute B−1B^{-1}B−1.

Since B=[1203],B=\begin{bmatrix}1&2\\0&3\end{bmatrix},B=[10​23​], its determinant is ∣B∣=1⋅3−0⋅2=3.|B|=1\cdot 3-0\cdot 2=3.∣B∣=1⋅3−0⋅2=3. Therefore, B−1=13[3−201]=[1−23013].B^{-1}=\frac{1}{3}\begin{bmatrix}3&-2\\0&1\end{bmatrix}=\begin{bmatrix}1&-\frac23\\0&\frac13\end{bmatrix}.B−1=31​[30​−21​]=[10​−32​31​​].

So, A=λ[1−23013].A=\lambda \begin{bmatrix}1&-\frac23\\0&\frac13\end{bmatrix}.A=λ[10​−32​31​​].

  1. Use the condition ∣3A∣=108|3A|=108∣3A∣=108.

For a 2×22\times 22×2 matrix, ∣3A∣=32∣A∣=9∣A∣.|3A|=3^2|A|=9|A|.∣3A∣=32∣A∣=9∣A∣. Thus, 9∣A∣=108  ⟹  ∣A∣=12.9|A|=108 \implies |A|=12.9∣A∣=108⟹∣A∣=12.

Now, ∣A∣=∣λB−1∣=λ2∣B−1∣.|A|=\left|\lambda B^{-1}\right|=\lambda^2|B^{-1}|.∣A∣=​λB−1​=λ2∣B−1∣. Since ∣B−1∣=1∣B∣=13,|B^{-1}|=\frac{1}{|B|}=\frac13,∣B−1∣=∣B∣1​=31​, we get λ2⋅13=12\lambda^2\cdot \frac13=12λ2⋅31​=12 λ2=36.\lambda^2=36.λ2=36.

Thus, A2=λ2(B−1)2=36(B−1)2.A^2=\lambda^2(B^{-1})^2=36(B^{-1})^2.A2=λ2(B−1)2=36(B−1)2.

  1. Compute (B−1)2(B^{-1})^2(B−1)2.

B−1=[1−23013]B^{-1}=\begin{bmatrix}1&-\frac23\\0&\frac13\end{bmatrix}B−1=[10​−32​31​​] so

(B−1)2=[1−23013][1−23013]=[1−23−29019]=[1−89019].(B^{-1})^2= \begin{bmatrix}1&-\frac23\\0&\frac13\end{bmatrix} \begin{bmatrix}1&-\frac23\\0&\frac13\end{bmatrix} = \begin{bmatrix} 1&-\frac23-\frac29\\[4pt] 0&\frac19 \end{bmatrix} = \begin{bmatrix} 1&-\frac89\\[4pt] 0&\frac19 \end{bmatrix}.(B−1)2=[10​−32​31​​][10​−32​31​​]=[10​−32​−92​91​​]=[10​−98​91​​].

Therefore,

=\begin{bmatrix}36&-32\\0&4\end{bmatrix}.$$ 5. Compare with the options. This matches $$\boxed{\begin{bmatrix}36&-32\\0&4\end{bmatrix}}$$ which is option **D**.
PreviousNext

More from Matrices and Determinants

  • If the system of linear equations x + ay + z = 3 x + 2y + 2z = 6 x + 5y + 3z = b has no solution, then :2018 · MCQ
  • Suppose A is any 3 × 3 non-singular matrix and ( A − 3I) (A − 5I) = O where I = I3 and O = O3. If α A + β A-1 = 4I, then α+β is equal to :2018 · MCQ
  • Let A = ​111​011​001​​ and B = A20. Then the sum of the elements of the first column of B is :2018 · MCQ
  • The number of values of k for which the system of linear equations, (k + 2)x + 10y = k kx + (k +3)y = k -1 has no solution, is :2018 · MCQ
  • If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then y2xz​ is equal to2018 · MCQ
  • If ​x−42x2x​2xx−42x​2x2xx−4​​=(A+Bx)(x−A)2 then the ordered pair (A, B) is…2018 · MCQ
  • The number of real values of λ for which the system of linear equations 2x + 4y −λ z = 0 4x + λ y + 2z = 0 λ x + 2y + 2z = 0 has infinitely many solutions, is :2017 · MCQ
  • Let A be any 3 × 3 invertible matrix. Then which one of the following is not always true ?2017 · MCQ