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NCERT Exemplar · Class 12 Mathematics Matrices

101 questions · 101 still being checked

EXERCISE 3.3 71–80 (part 8 of 10)

  1. Fill in the blanks in each of the Exercises $\displaystyle 68$–81.

    Exercise 71

    The product of any matrix by the scalar ____ is the null matrix.

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    NCERT’s answer
    $\displaystyle 0$
    \(\displaystyle 0\)\[0\cdot[a_{ij}]=[0\cdot a_{ij}]=[0]=O \]

  2. Exercise 72

    A matrix which is not a square matrix is called a ____ matrix.

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    Rectangular matrix
    RectangularA matrix of order \(\displaystyle m\times n\) with \(\displaystyle m\neq n\) has unequal numbers of rows and columns.

  3. Exercise 73

    Matrix multiplication is ____ over addition.

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    NCERT’s answer
    Distributive
    distributive\[A(B+C) = AB + AC \] \[(A+B)C = AC + BC \]Both hold whenever the products and sums are defined.

  4. Exercise 74

    If A is a symmetric matrix, then A3\displaystyle \mathrm{A}^3 is a ____ matrix.

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    NCERT’s answer
    Symmetrix matrix
    symmetric\[A' = A \quad \text{(given)} \] \[(A^3)' = (A')^3 = A^3 \]Since \(\displaystyle (A^3)' = A^3\), \(\displaystyle A^3\) is symmetric.

  5. Exercise 75

    If A is a skew symmetric matrix, then A2\displaystyle \mathrm{A}^2 is a ____.

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    NCERT’s answer
    Symmetrix matrix
    symmetric matrix\[A' = -A \quad \text{(given)} \] \[(A^2)' = (A')^2 = (-A)^2 = A^2 \]Since \(\displaystyle (A^2)' = A^2\), \(\displaystyle A^2\) is symmetric.

  6. Exercise 76

    If A and B are square matrices of the same order, then
    (i)
    (AB)′=\displaystyle (\mathrm{AB})^{\prime}= ____.
    (ii)
    (kA)′=\displaystyle (k \mathrm{A})^{\prime}= ____. (k\displaystyle k is any scalar)
    (iii)
    [k(A−B)]′=\displaystyle [k(\mathrm{A}-\mathrm{B})]^{\prime}= ____.

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    NCERT’s answer
    (i)
    \(\displaystyle \mathrm{B}^{\prime} \mathrm{A}^{\prime}\)
    (ii)
    \(\displaystyle k \mathrm{A}^{\prime}\)
    (iii)
    \(\displaystyle k(\mathrm{A}^{\prime}-\mathrm{B}^{\prime})\)
    (i) \(\displaystyle B'A'\) (ii) \(\displaystyle kA'\) (iii) \(\displaystyle k(A'-B')\)\[(AB)' = B'A' \quad \text{(reversal law)} \] \[(kA)' = kA' \] \[[k(A-B)]' = k(A-B)' = k(A'-B') \]

  7. Exercise 77

    If A is skew symmetric, then kA\displaystyle k \mathrm{A} is a ____. (k\displaystyle k is any scalar)

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    NCERT’s answer
    Skew Symmetric matrix
    skew symmetric matrix\[A' = -A \quad \text{(given)} \] \[(kA)' = kA' = k(-A) = -(kA) \]Since \(\displaystyle (kA)' = -(kA)\), \(\displaystyle kA\) is skew symmetric.

  8. Exercise 78

    If A and B are symmetric matrices, then
    (i)
    AB−BA\displaystyle \mathrm{AB}-\mathrm{BA} is a ____.
    (ii)
    BA−2AB\displaystyle \mathrm{BA}-2 \mathrm{AB} is a ____.

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    NCERT’s answer
    (i)
    Skew symmetric matrix
    (ii)
    neither symmetric nor skew symmetric matrix
    (i) skew symmetric (ii) neither symmetric nor skew symmetric\[A' = A, \qquad B' = B \quad \text{(given)} \](i) \[(AB-BA)' = (AB)' - (BA)' = B'A' - A'B' = BA - AB = -(AB-BA) \](ii) \[(BA-2AB)' = A'B' - 2B'A' = AB - 2BA \]This equals \(\displaystyle BA-2AB\) only if \(\displaystyle AB=BA\), and equals \(\displaystyle -(BA-2AB)\) only if \(\displaystyle AB=-BA\). In general neither holds.

  9. Exercise 79

    If A is symmetric matrix, then B′AB\displaystyle \mathrm{B}^{\prime} \mathrm{AB} is ____.

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    NCERT’s answer
    Symmetric matrix
    symmetric\[A' = A \quad \text{(given)} \] \[(B'AB)' = B'A'(B')' = B'AB \]Since \(\displaystyle (B'AB)' = B'AB\), \(\displaystyle B'AB\) is symmetric.

  10. Exercise 80

    If A and B are symmetric matrices of same order, then AB is symmetric if and only if ____.

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    NCERT’s answer
    \(\displaystyle \mathrm{AB}=\mathrm{BA}\)
    \(\displaystyle AB = BA\)\[A' = A, \qquad B' = B \quad \text{(given)} \] \[(AB)' = B'A' = BA \] \[AB \text{ symmetric} \iff (AB)' = AB \iff BA = AB \]