Q.If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, then verify that

Showing the 12 most recent of 22 on this concept.

1. CBSE 2026Set 65/1/11 markMCQ
   Q.Which of the following properties is/are true for two matrices of suitable orders?

   (i) $(A + B)' = A' + B'$

   (ii) $(A - B)' = B' - A'$

   (iii) $(AB)' = A'B'$

   (iv) $(kAB)' = kB'A'$ ($k$ is a scalar) (A)

   (i) only (B) (i),

   (ii) and

   (iii) (C)

   (i) and

   (ii) (D)

   (i) and (iv)

   ›Reveal solutionSolution

   The transpose of a sum is the sum of transposes, and the transpose of a product reverses the order. Only statements (i) and (iv) are correct.

   The transpose operation flips a matrix over its diagonal — rows become columns and columns become rows. The key intuition is that transposition distributes over addition but reverses the order of multiplication. This reversal is not arbitrary; it comes from the fact that when you multiply two matrices and then transpose, the dimensions must still match, which forces the order swap.

   Let’s check each statement carefully.

   1. Statement (i): $(A + B)' = A' + B'$

      This is true. Transposition is a linear operation — adding two matrices and then transposing gives the same result as transposing each first and then adding. Element-wise, the $(i,j)$ entry of $(A+B)'$ is $a_{ji} + b_{ji}$, which is exactly the $(i,j)$ entry of $A' + B'$.

   2. Statement (ii): $(A - B)' = B' - A'$

      This is false. The correct property is $(A - B)' = A' - B'$, because transposition distributes over subtraction just as it does over addition. The given expression has the order swapped, which is wrong. For example, take $A = \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$ and $B = \begin{pmatrix}0 & 0 \\ 1 & 0\end{pmatrix}$; the left side gives $\begin{pmatrix}1 & -1 \\ 0 & 0\end{pmatrix}' = \begin{pmatrix}1 & 0 \\ -1 & 0\end{pmatrix}$, while the right side gives $\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix} - \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix} = \begin{pmatrix}-1 & 1 \\ 0 & 0\end{pmatrix}$, which are not equal.

   3. Statement (iii): $(AB)' = A'B'$

      This is false. The correct property is $(AB)' = B'A'$ — the order of multiplication reverses. The reason is dimensional: if $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$, so $(AB)'$ is $p \times m$. For $A'B'$ to be defined, $A'$ would need to be $n \times m$ and $B'$ would be $p \times n$, which cannot multiply in that order unless $m = p$. The correct product $B'A'$ has $B'$ as $p \times n$ and $A'$ as $n \times m$, giving a $p \times m$ result — matching dimensions perfectly.

   Watch out

   A common mistake is to forget the reversal in the transpose of a product. Always remember: the transpose of a product is the product of the transposes in reverse order.

   4. Statement (iv): $(kAB)' = kB'A'$ This is true. The scalar $k$ is a constant, so it factors out unchanged: $(kAB)' = k(AB)' = k(B'A')$. The order reversal is the same as in statement (iii), and the scalar simply tags along.

   Tip

   You can remember the reversal rule by thinking of socks and shoes: you put on socks then shoes, but to undo (transpose) you take off shoes first then socks — the order reverses.

   Only statements (i) and (iv) are correct.

   ✓Final answer

   The correct option is (D) (i) and (iv).

2. CBSE 2020Set 65/1/11 mark
   Q.If the order of matrix A is $3 \times 2$, then the order of matrix A' will be _________.

   (OR)

   A square matrix A will be a skew-symmetric matrix, if _________.

   ›Reveal solutionSolution

   1. $A'$ has order $2\times3$.
   2. $A$ is skew-symmetric iff $A'=-A$.

   Part (a)

   The transpose $A'$ of a matrix is formed by interchanging its rows and columns: an entry in row $i$, column $j$ of $A$ moves to row $j$, column $i$ of $A'$. So an $m\times n$ matrix has an $n\times m$ transpose. With $A$ of order $3\times2$ ($m=3,n=2$):

   $\text{order of }A'=2\times3.$

   ✓Final answer

   The order of $A'$ is $2\times 3$.

   Part (b)

   A square matrix $A$ is skew-symmetric (anti-symmetric) precisely when its transpose equals its negative:

   $A'=-A,\qquad\text{i.e. } a_{ji}=-a_{ij}\ \text{for all }i,j.$

   Setting $i=j$ gives $a_{ii}=-a_{ii}$, so every diagonal entry is $0$.

   ✓Final answer

   $A$ is skew-symmetric if $A'=-A$ (its transpose equals its negative).

3. CBSE 2026Set 65/2/11 markMCQ
   Q.If $A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}$ and $A + A' = I$, then the value of $x \in \left[0, \frac{\pi}{2}\right]$ is (A) $0$ (B) $\frac{\pi}{4}$ (C) $\frac{\pi}{3}$ (D) $\frac{\pi}{2}$

   ›Reveal solutionSolution

   The key idea is that $A$ is a rotation matrix, $A'$ is its transpose (which is also its inverse), and the condition $A + A' = I$ forces the diagonal sum $2\cos x = 1$, giving $x = \frac{\pi}{3}$.

   We are given a $2 \times 2$ matrix $A$ that depends on an angle $x$. The matrix $A$ is a classic rotation matrix: it rotates a vector in the plane by angle $x$ counterclockwise. Its transpose $A'$ is simply the rotation by $-x$ (clockwise), which is also the inverse of $A$.

   The condition $A + A' = I$ means that when we add the matrix and its transpose, we get the identity matrix. This is a direct equation in the entries of the matrices.

   Let’s write it out step by step.

   1. Write $A$ and $A'$ explicitly.

      $A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}$.

      The transpose $A'$ swaps rows and columns:

      $A' = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix}$.

   2. Add them entrywise.

      $A + A' = \begin{bmatrix} \cos x + \cos x & -\sin x + \sin x \\ \sin x + (-\sin x) & \cos x + \cos x \end{bmatrix} = \begin{bmatrix} 2\cos x & 0 \\ 0 & 2\cos x \end{bmatrix}$.

      Notice the off-diagonal terms cancel perfectly: $-\sin x + \sin x = 0$ and $\sin x - \sin x = 0$. So the sum is a diagonal matrix with both diagonal entries equal to $2\cos x$.

   3. Set this equal to $I$.

      The identity matrix $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

      So we require:

      $\begin{bmatrix} 2\cos x & 0 \\ 0 & 2\cos x \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

      This gives a single equation from the diagonal entries: $2\cos x = 1$.

   4. Solve for $x$ in the given interval.

      $2\cos x = 1 \implies \cos x = \frac{1}{2}$.

      The interval is $x \in \left[0, \frac{\pi}{2}\right]$. The angle whose cosine is $\frac{1}{2}$ in this interval is $x = \frac{\pi}{3}$ (since $\cos \frac{\pi}{3} = \frac{1}{2}$).

      The other standard angle $\frac{\pi}{6}$ gives $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, not $\frac{1}{2}$.

   Watch out

   A common mistake is to forget that $A'$ is the transpose, not the conjugate transpose. Here all entries are real, so it's fine — but also note that $A'$ is not the same as $A^{-1}$ in general; it just happens to be here because $A$ is orthogonal.

   Tip

   Recognising $A$ as a rotation matrix immediately tells you that $A + A' = 2\cos x \cdot I$, because the sum of a rotation and its inverse (transpose) is always a scalar multiple of the identity. This saves you from writing out all four entries.

   ✓Final answer

   The value of $x$ is $\boxed{\frac{\pi}{3}}$, which corresponds to option (C).

4. CBSE 2026Set A1 markMCQ
   Q.$A = [4\ 2\ 3] \Rightarrow A' =$

   (a) $[4\ 2\ 3]$

   (b) $\begin{bmatrix} 3 \\ 2 \\ 4 \end{bmatrix}$

   (c) $[3\ 2\ 4]$

   (d) $\begin{bmatrix} 4 \\ 2 \\ 3 \end{bmatrix}$

   ›Reveal solutionSolution

   Transpose turns the row $[4\ 2\ 3]$ into the column $\begin{bmatrix}4\\2\\3\end{bmatrix}$.

   The transpose $A'$ (or $A^T$) interchanges rows and columns, keeping each entry's value. A $1\times3$ row matrix becomes a $3\times1$ column matrix with the entries listed top-to-bottom in the same order:

   $A=[4\ 2\ 3]\Rightarrow A'=\begin{bmatrix}4\\2\\3\end{bmatrix}$.

   ✓Final answer

   (D) $\begin{bmatrix}4\\2\\3\end{bmatrix}$.

5. CBSE 2026Set ANNUAL1 markMCQ
   Q.If the order of the matrix A is 2×3 then the order of the matrix (A')' is:

   (a) 2×3

   (b) 3×2

   (c) 2×2

   (d) 3×3

   ›Reveal solutionSolution

   The transpose of the transpose of a matrix is the matrix itself, so $(A')' = A$, which has the same order as $A$.

   For any matrix $A$, the property $(A')' = A$ always holds (taking the transpose twice returns the original matrix).

   Since $A$ has order $2\times3$, $(A')'$ also has order $2\times3$.

   ✓Final answer

   The order of $(A')'$ is $2\times3$ (option a).

6. CBSE 2026Set ANNUAL1 markMCQ
   Q.If A = [[cos α, −sin α], [sin α, cos α]], then A + A' = I if the value of α is

   (a) π/6

   (b) π/3

   (c) π

   (d) 3π/2

   ›Reveal solutionSolution

   Adding $A$ to its transpose cancels the sine terms and leaves a diagonal matrix of $2\cos\alpha$; setting this equal to $I$ pins down $\alpha$.

   $A = \begin{pmatrix}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha\end{pmatrix}$, so $A' = \begin{pmatrix}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha\end{pmatrix}$.

   $A+A' = \begin{pmatrix}2\cos\alpha & 0\\ 0 & 2\cos\alpha\end{pmatrix}$

   For this to equal $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$, we need $2\cos\alpha = 1 \Rightarrow \cos\alpha = \dfrac12 \Rightarrow \alpha = \dfrac{\pi}{3}$.

   ✓Final answer

   $\alpha = \pi/3$. (Option b)

7. CBSE 2025Set E1 markMCQ
   Q.$A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \Rightarrow A' =$

   (a) $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

   (b) $\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$

   (c) $\begin{bmatrix} 3 & 2 & 1 \end{bmatrix}$

   (d) $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$

   ›Reveal solutionSolution

   Transposing the row $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ gives the column $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$.

   The transpose $A'$ turns rows into columns while keeping the entries in the same order. So the $1\times3$ matrix $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ becomes the $3\times1$ matrix

   $A' = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}.$

   Note (B) reverses the order, so it is wrong.

   ✓Final answer

   (D) $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$.

8. CBSE 2025Set A1 markMCQ
   Q.If $A = \begin{bmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{bmatrix}$ and $A + A' = I$, then the value of $\alpha$ is:

   (a) $\dfrac{\pi}{6}$

   (b) $\pi$

   (c) $\dfrac{3\pi}{2}$

   (d) $\dfrac{\pi}{3}$

   ›Reveal solutionSolution

   Compute $A'$ (transpose), add to $A$, and match to the identity matrix.

   $A = \begin{bmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{bmatrix}, \qquad A' = \begin{bmatrix}\cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha\end{bmatrix}$

   $A + A' = \begin{bmatrix}2\cos\alpha & 0 \\ 0 & 2\cos\alpha\end{bmatrix}$

   We are given $A + A' = I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. Comparing entries:

   $2\cos\alpha = 1 \implies \cos\alpha = \dfrac12 \implies \alpha = \dfrac{\pi}{3}$

   ✓Final answer

   (d) $\dfrac{\pi}{3}$.

9. CBSE 2025Set ANNUAL1 mark
   Q.If order of the matrix A is 3×2 then order of matrix (A')' is ______.

   ›Reveal solutionSolution

   Taking the transpose twice returns the original matrix, so its order is unchanged.

   For any matrix $A$, $(A')' = A$ (transposing twice undoes the operation). Since $A$ has order $3\times2$, so does $(A')'$.

   ✓Final answer

   $3\times2$.

10. CBSE 2024Set D1 markMCQ
    Q.If $A = \begin{bmatrix} 2 & \sqrt{2} & 0 \\ 3 & -2 & \frac{2}{5} \end{bmatrix}$ then $A' =$

    (a) $\begin{bmatrix} 2 & -3 \\ \sqrt{2} & 2 \\ 0 & 2/5 \end{bmatrix}$

    (b) $\begin{bmatrix} 2 & 3 \\ \sqrt{2} & -2 \\ 0 & 2/5 \end{bmatrix}$

    (c) $\begin{bmatrix} 3 & 2 \\ -2 & \sqrt{2} \\ -2/5 & 0 \end{bmatrix}$

    (d) $\begin{bmatrix} 3 & -2 & 2/5 \\ 2 & \sqrt{2} & 0 \end{bmatrix}$

    ›Reveal solutionSolution

    The transpose $A'$ turns each row of $A$ into a column.

    $A = \begin{bmatrix} 2 & \sqrt{2} & 0 \\ 3 & -2 & \frac{2}{5} \end{bmatrix}$ is $2\times3$, so $A'$ is $3\times2$ with columns equal to the rows of $A$:

    $A' = \begin{bmatrix} 2 & 3 \\ \sqrt{2} & -2 \\ 0 & 2/5 \end{bmatrix}.$

    ✓Final answer

    (B) $\begin{bmatrix} 2 & 3 \\ \sqrt{2} & -2 \\ 0 & 2/5 \end{bmatrix}$

11. CBSE 2024Set ANNUAL1 markMCQ
    Q.If $A$ is a matrix of order $2 \times 3$ and $B$ is a matrix of order $3 \times 4$, then the order of $(AB)'$ is

    (a) $2 \times 3$

    (b) $2 \times 4$

    (c) $4 \times 2$

    (d) $3 \times 4$

    ›Reveal solutionSolution

    Order of $AB$ is (rows of $A$) $\times$ (columns of $B$); a transpose swaps rows and columns.

    $A$ is $2\times 3$ and $B$ is $3\times 4$. Since the number of columns of $A$ (=3) equals the number of rows of $B$ (=3), $AB$ is defined and has order (rows of $A$) $\times$ (columns of $B$) $= 2\times 4$.

    The transpose of a matrix of order $m\times n$ has order $n\times m$. So $(AB)'$ has order $4\times 2$.

    ✓Final answer

    Order of $(AB)' = 4\times 2$ (option c).

12. CBSE 2024Set ANNUAL1 mark
    Q.If $A = \begin{bmatrix}1\\2\\4\end{bmatrix}_{3\times1}$ and $B = \begin{bmatrix}2 & 0 & 1\end{bmatrix}_{1\times3}$, then find the matrix (AB)', where (AB)' is the transpose of matrix (AB).

    ›Reveal solutionSolution

    First multiply A (3×1) and B (1×3) to get a 3×3 matrix, then transpose it.

    Given $A = \begin{bmatrix}1\\2\\4\end{bmatrix}_{3\times1}$ and $B = \begin{bmatrix}2 & 0 & 1\end{bmatrix}_{1\times3}$.

    Step 1: Compute AB (3×3).

    $AB = \begin{bmatrix}1\\2\\4\end{bmatrix}\begin{bmatrix}2 & 0 & 1\end{bmatrix} = \begin{bmatrix}1(2) & 1(0) & 1(1)\\ 2(2) & 2(0) & 2(1)\\ 4(2) & 4(0) & 4(1)\end{bmatrix} = \begin{bmatrix}2 & 0 & 1\\ 4 & 0 & 2\\ 8 & 0 & 4\end{bmatrix}$

    Step 2: Transpose AB (rows become columns).

    $(AB)' = \begin{bmatrix}2 & 4 & 8\\ 0 & 0 & 0\\ 1 & 2 & 4\end{bmatrix}$

    ✓Final answer

    $(AB)' = \begin{bmatrix}2 & 4 & 8\\ 0 & 0 & 0\\ 1 & 2 & 4\end{bmatrix}$.