SolveIt is under development
SolveItNCERT · CBSE · NEET

NCERT Exemplar · Class 12 Mathematics Matrices

101 questions · 101 still being checked

EXERCISE 3.3 81–90 (part 9 of 10)

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

    Exercise 81

    In applying one or more row operations while finding A−1\displaystyle \mathrm{A}^{-1} by elementary row operations, we obtain all zeros in one or more, then A−1\displaystyle \mathrm{A}^{-1} ____.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    does not exist
    does not exist. A zero row in the reduced left-hand matrix forces \(\displaystyle |A| = 0\).\[E_k \cdots E_2 E_1 A = B \quad \text{(} B \text{ has a zero row)} \]\[|B| = 0 \]\[|E_k| \cdots |E_1|\,|A| = |B| = 0, \quad |E_i| \neq 0 \]\[|A| = 0 \;\Rightarrow\; A \text{ is singular} \]
  2. State Exercises $\displaystyle 82$ to $\displaystyle 101$ which of the following statements are True or False

    Exercise 82

    A matrix denotes a number.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    False
    False. A matrix is a rectangular array of numbers; only its determinant is a single number.\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad \text{(4 entries, order } 2 \times 2 \text{)} \]\[|A| = 1 \cdot 4 - 2 \cdot 3 = -2 \quad \text{(a number)} \]

  3. Exercise 83

    Matrices of any order can be added.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    False
    False. Addition is entrywise, so both matrices must have the same order.\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}_{2 \times 2}, \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}_{2 \times 3} \]\[A + B \quad \text{is not defined (3 and 6 have no partner in } A \text{)} \]

  4. Exercise 84

    Two matrices are equal if they have same number of rows and same number of columns.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    False
    False. Equal matrices need the same order and equal corresponding entries.\[\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \neq \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix} \quad \text{(both } 2 \times 2 \text{)} \]

  5. Exercise 85

    Matrices of different order can not be subtracted.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    True
    True. Subtraction is entrywise, so it needs the same order, as addition does.\[A - B = A + (-B) \]\[(A - B)_{ij} = a_{ij} - b_{ij} \quad \text{(needs } b_{ij} \text{ for every } a_{ij} \text{)} \]

  6. Exercise 86

    Matrix addition is associative as well as commutative.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    True
    True. Both laws follow entrywise from the same laws for real numbers.\[(A+B)_{ij} = a_{ij} + b_{ij} = b_{ij} + a_{ij} = (B+A)_{ij} \]\[\big((A+B)+C\big)_{ij} = (a_{ij} + b_{ij}) + c_{ij} = a_{ij} + (b_{ij} + c_{ij}) = \big(A+(B+C)\big)_{ij} \]

  7. Exercise 87

    Matrix multiplication is commutative.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    False
    False. Counterexample:\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]\[AB = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix} \]\[BA = \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix} \neq AB \]

  8. Exercise 88

    A square matrix where every element is unity is called an identity matrix.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    False
    False. An identity matrix has $\displaystyle 1$ on the diagonal and $\displaystyle 0$ elsewhere, and satisfies \(\displaystyle I^2 = I\). The all-ones matrix does not:\[J = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \]\[J^2 = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} \neq J \]

  9. Exercise 89

    If A and B are two square matrices of the same order, then A+B=B+A\displaystyle \mathrm{A}+\mathrm{B}=\mathrm{B}+\mathrm{A}.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    True
    True. Matrices add entry by entry, and addition of numbers is commutative. \[(A+B)_{ij} = a_{ij} + b_{ij} \] \[a_{ij} + b_{ij} = b_{ij} + a_{ij} = (B+A)_{ij} \] \[\therefore\; A + B = B + A \]

  10. Exercise 90

    If A and B are two matrices of the same order, then A−B=B−A\displaystyle \mathrm{A}-\mathrm{B}=\mathrm{B}-\mathrm{A}.

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    False
    False. The equality forces \(\displaystyle A = B\), which is not true in general. \[A - B = B - A \;\Rightarrow\; 2A = 2B \;\Rightarrow\; A = B \] Counter-example: \[A=\begin{bmatrix}1&0\\0&1\end{bmatrix},\quad B=\begin{bmatrix}0&0\\0&0\end{bmatrix} \] \[A-B=\begin{bmatrix}1&0\\0&1\end{bmatrix} \neq \begin{bmatrix}-1&0\\0&-1\end{bmatrix}=B-A \]