PYQ Vault
EbooksTutor

Find a matrix, an entry or its determinant from its products with given column vectors

This type of question, asked 6 times: 27 Jan 2024, 23 Jan 2025, 24 Jan 2025, 22 Jan 2026 S1, 5 Apr 2026 S1, 5 Apr 2026 S2. All 6 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    27 Jan 2024

    89
    Let A=[201110101],B=[B1, B2, B3]A =\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix},B =\left\lbrack B_{1},{\text{ }B}_{2},{\text{ }B}_{3} \right\rbrack, where B1B_{1}, B2, B3B_{2},{\text{ }B}_{3} are column matrices, and AB1=[100]AB_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},
    AB2=[230],AB3=[321]AB_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix},AB_{3}=\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}
    If α=∣B∣\alpha = |B| and β\beta is the sum of all the diagonal elements of BB, then α3+β3\alpha^{3}+\beta^{3} is equal to .

  2. Question 2

    23 Jan 2025

    141
    Let A=[aij]A =\left\lbrack a_{ij} \right\rbrack be a 3×33 \times 3 matrix such that A[010]=[001],A[413]=[010]A\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix},A\begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} and A[212]=[100]A\begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, then a23a_{23} equals:

    Tap an option to check your answer.

  3. Question 3

    24 Jan 2025

    74
    Let be a 3×33 \times 3 matrix such that XTAX=OX^{T}AX = O for all nonzero 3×13 \times 1 matrices X=[xyz]X =\begin{bmatrix} x \\ y \\ z \end{bmatrix}. If A[111]=[14−5],A[121]=[04−8]A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=\begin{bmatrix} 1 \\ 4 \\ - 5 \end{bmatrix},A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=\begin{bmatrix} 0 \\ 4 \\ - 8 \end{bmatrix}, and det(adj(2( A+I)))=2α3β5γ,α,β,γ,∈Ndet(adj(2(\text{ }A + I))) =2^{\alpha}3^{\beta}5^{\gamma},\alpha,\beta,\gamma, \in N, then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is

  4. Question 4

    22 Jan 2026 S1

    72
    Let A be a 3×33 \times 3 matrix such that A+AT=OA + A^{T} = O. If A[1−10]=[332],A2[1−10]=[−319−24]A\begin{bmatrix} 1 \\ - 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix},A^{2}\begin{bmatrix} 1 \\ - 1 \\ 0 \end{bmatrix} = \begin{bmatrix} - 3 \\ 19 \\ - 24 \end{bmatrix} and det(adj(2adj(A+I)))=(2)α⋅(3)β⋅(11)γ,α,β,γdet(adj(2adj(A + I))) = (2)^{\alpha} \cdot (3)^{\beta} \cdot (11)^{\gamma},\alpha,\beta,\gamma are non-negative integers, then α+β+γ\alpha + \beta + \gamma is equal to ____\_\_\_\_

  5. Question 5

    5 Apr 2026 S1

    53
    Let A be a 3×33 \times 3 matrix such that AT[101]=[522], AT[001]=[311], A[101]=[344]A^{T}\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}=\begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix},\ A^{T}\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}=\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix},\ A\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}=\begin{bmatrix} 3 \\ 4 \\ 4 \end{bmatrix} and A[001]=[131]A\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} If det(A)=1det(A) = 1, then det⁡(adj(A2+A))\det\left( adj\left( A^{2}+ A \right) \right) is equal to :

    Tap an option to check your answer.

  6. Question 6

    5 Apr 2026 S2

    54
    Let M be a 3×33 \times 3 matrix such that M(100)=(123),M(010)=(012)M\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},M\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} and M(001)=(−111)M\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}=\begin{pmatrix} - 1 \\ 1 \\ 1 \end{pmatrix}. If M(xyz)=(1711)M\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}, then x+y+zx + y + z equals :

    Tap an option to check your answer.