Let \( A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \). If \( A + B + C = O \), then matrix \( C \) is :
Show Hint
- \( \begin{bmatrix} -3 & -3 & 5\\ 3 & 4 & 2 \end{bmatrix} \)
- \( \begin{bmatrix} 3 & 3 & 5 \\ -3 & -4 & -2 \end{bmatrix} \)
- \( \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} \)
- \( \begin{bmatrix} -3 & -3 & -5 \\ 3 & 4 & 2 \end{bmatrix} \)
The Correct Option is C
Solution and Explanation
Concept:
Matrix addition is performed by adding the corresponding elements of the matrices.
If \[ A+B+C=O, \] then \[ C=-(A+B). \]
Step 1: Calculate \(A+B\)
Given: \[ A=\begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix}, \quad B=\begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \] Adding corresponding elements: \[ A+B= \begin{bmatrix} 0+(-3) & -3+0 & 4+1 \\ 1+2 & 0+4 & 2+0 \end{bmatrix} \] \[ A+B= \begin{bmatrix} -3 & -3 & 5 \\ 3 & 4 & 2 \end{bmatrix} \]
Step 2: Find matrix \(C\)
Since \[ A+B+C=O, \] we get: \[ C=-(A+B) \] Therefore, \[ C=- \begin{bmatrix} -3 & -3 & 5 \\ 3 & 4 & 2 \end{bmatrix} \]
Step 3: Negate each element
\[ C= \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} \]
Final Answer:
\[ \boxed{ C= \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} } \]
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Operations on Matrices Questions
- If $A$ and $B$ are square matrices of the same order, then which of the following statements is/are always true?
(i) $(A + B)(A - B) = A^2 - B^2$
(ii) $AB = BA$
(iii) $(A + B)^2 = A^2 + AB + BA + B^2$
(iv) $AB = 0 \implies A = 0$ or $B = 0$ - Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative. - If \( A \) and \( B \) are square matrices of same order, then which of the following statements is/are always true?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B)^2=A^2+AB+BA+B^2$
(iv) $AB=0\Rightarrow A=0\text{ or }B=0$}
- If a square matrix \(A\) is such that \(A^2 = A\) and \((I - A)^3 = xA + I\), then value of \(x\) must be :
- Given that $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, $Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}$ and $R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}$, find a matrix $S$ such that $PQ - RS$ is a null matrix.}
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is