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Federal (FBISE) · Chapter test · Free

1st Year Mathematics Chapter 2 Test2. Matrices and Determinants

30 marks · 40 minutes · 21 questions. Covers: 2. Matrices and Determinants.

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1st Year Mathematics Chapter 2 Test

1st Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 30Time: 40 minutes

Q1. Multiple choice questions (Attempt all · 10 × 1 = 10)

  1. For the matrices A=[23−51] A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix} and B=[4−113] B = \begin{bmatrix} 4 & -1 \\ 1 & 3 \end{bmatrix} , the value of matrix X in the matrix equations 2X - A = B will be :
    1. [124−88] \begin{bmatrix} 12 & 4 \\ -8 & 8 \end{bmatrix}
    2. [62−44] \begin{bmatrix} 6 & 2 \\ -4 & 4 \end{bmatrix}
    3. [31−22] \begin{bmatrix} 3 & 1 \\ -2 & 2 \end{bmatrix}
    4. [93−33] \begin{bmatrix} 9 & 3 \\ -3 & 3 \end{bmatrix}
  2. The matrix in reduced echelon form is :
    1. [01−120610234−500010.401100]\begin{bmatrix} 0 & 1 & -\frac{1}{2} & 0 & 6 \\ 1 & 0 & \frac{2}{3} & 4 & -5 \\ 0 & 0 & 0 & 1 & 0.4 \\ 0 & 1 & 1 & 0 & 0 \end{bmatrix}
    2. [13−61000014−300011200000]\begin{bmatrix} 1 & 3 & -6 & 10 & 0 \\ 0 & 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}
    3. [130000014120010−300000]\begin{bmatrix} 1 & 3 & 0 & 0 & 0\\ 0 & 0 & 1 & 4 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 0 & 0\end{bmatrix}
    4. [130000010−300011200000]\begin{bmatrix} 1 & 3 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}
  3. The principal diagonal / main diagonal elements in the matrix A=[1−3124−1356] A=\begin{bmatrix} 1 & -3 & 1 \\ 2 & 4 & -1 \\ 3 & 5 & 6 \end{bmatrix} are:
    1. 1 , - 3 , 1
    2. 1 , 4 , 3
    3. 1 , 4 , 6
    4. 3 , 5 , 6
  4. If A=[123−3] A = \begin{bmatrix} 1 & 2 \\ 3 & - 3 \end{bmatrix} and B=[−16] B= \begin{bmatrix} - 1 \\ 6 \end{bmatrix} , given that AX =B , then X =
    1. [11] \begin{bmatrix} 1 \\ 1 \end{bmatrix}
    2. [−11] \begin{bmatrix} -1 \\ 1 \end{bmatrix}
    3. [1−1] \begin{bmatrix} 1 \\ -1 \end{bmatrix}
    4. [−1−1] \begin{bmatrix} -1 \\ -1 \end{bmatrix}
  5. If A and B are two matrices and O ( A + B ) = 2×32 \times3 , then O ( B A ) =
    1. 2×22 \times 2
    2. 3×33\times 3
    3. 3×23\times 2
    4. Not possible
  6. For a system of non-homogeneous equations with three variables system will have unique solution if.
    1. Rank A < 3
    2. Rank AbA_{b} < 3
    3. Rank A = Rank AbA_{b} = 3
    4. Rank A = Rank AbA_{b} < 3
  7. The number of non - zero rows of a matrix in the echelon form is called :
    1. Adjoint of a matrix
    2. Inverse of a matrix
    3. Rank of the matrix
    4. Transpose of a matrix
  8. The matrix [100020004]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} is a:
    1. Symmetric matrix
    2. Skew- symmetric matrix
    3. Identity matrix
    4. Null matrix
  9. For the matrix , A=[2−3560415−7], M13+A22= A = \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}, \, M_{13} + A_{22} =
    1. 30
    2. -19
    3. -49
    4. 11
  10. Suppose P and Q are two different matrices of order 3×n3 \times n and n×pn\times p respectively , then order of the matrix PQ will be :
    1. p×3p \times 3
    2. 3×p3 \times p
    3. n×nn \times n
    4. PQ is not defined

Q2. Short questions (Attempt any 6 of 8 · 6 × 2 = 12)

  1. Find the value of λ, so that the given matrix are singular.[λ20213λ21]\begin{bmatrix} \lambda & 2 & 0 \\ 2 & 1 & 3 \\ \lambda & 2 & 1 \end{bmatrix}
  2. If A=[120030216] A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 2 & 1 & 6 \end{bmatrix} then find ∣A∣=∣At∣ |A| = |A^t|
  3. If A=[123210020] A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 0 \\ 0 & 2 & 0 \end{bmatrix} and B=[123020210] B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix} then find ∣B∣=−∣A∣ |B| = -|A|
  4. Determine which of the following matrix are singular and which are non - singular.[312231−41−3] \begin{bmatrix} 3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3 \end{bmatrix}
  5. Identify the diagonal matrix, scalar matrix, identity matrix, lower triangular matrix, upper triangular matrix.C=[1020] C = \begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix}
  6. Construct a matrix B = [ aᵢⱼ ] of order 3 × 3 for which.bij=22i+j b_{ij} = \frac{2}{2i + j}
  7. Find the transpose of the following matrix and identify which one of them are symmetric and which are skew-symmetric.E=[3−69−620900] E = \begin{bmatrix} 3 & -6 & 9 \\ -6 & 2 & 0 \\ 9 & 0 & 0 \end{bmatrix}
  8. Find the multiplicative inverse of the following matrix if it exist by adjoint method.[1−1121−11−2−1] \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1 \end{bmatrix}

Q3. Long questions (Attempt any 2 of 3 · 2 × 4 = 8)

  1. Using the properties of the determinant prove the following.∣3a12a+12a+11a+2312∣=(a−1)(a−2) \begin{vmatrix} 3a & 1 & 2a+1 \\ 2a+1 & 1 & a+2 \\ 3 & 1 & 2 \end{vmatrix} = (a-1)(a-2)
  2. Solve the following system of linear equation by Gauss elimination method.2x+3y+4z=2;2x+y+z=5;3x−2y+z=−32x + 3y + 4z = 2 ; 2x + y + z = 5 ; 3x - 2y + z = -3
  3. Solve the following system of linear equation by Gauss-Jordan method.2x1−x2−x3=2;3x1−4x2+3x3=7;4x1+2x2−5x3=10 2x_1 - x_2 - x_3 = 2 ; 3x_1 - 4x_2 + 3x_3 = 7 ; 4x_1 + 2x_2 - 5x_3 = 10