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If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar) - Mathematics

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प्रश्न

If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar)

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उत्तर

If A and B are square matrices of the same order, then (kA)′ = kA'. (k is any scalar)

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पाठ 3: Matrices - Exercise [पृष्ठ ६३]

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एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12
पाठ 3 Matrices
Exercise | Q 76.(ii) | पृष्ठ ६३

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संबंधित प्रश्‍न

Show that AB ≠ BA in each of the following cases:

`A=[[1       3         0],[1        1          0],[4         1         0]]`And    B=`[[0      1          0],[1        0        0],[0           5          1]]`


Show that AB ≠ BA in each of the following cases:

`A = [[1,3,-1],[2,-1,-1],[3,0,-1]]` And `B= [[-2,3,-1],[-1,2,-1],[-6,9,-4]]`

 


Evaluate the following:

`[[],[1  2  3],[]]` `[[1     0      2],[2       0         1],[0          1       2]]` `[[2],[4],[6]]`


If A = `[[1     0],[0        1]]`,B`[[1            0],[0       -1]]`

and C= `[[0      1],[1       0]]` 

, then show that A2 = B2 = C2 = I2.

 

For the following matrices verify the associativity of matrix multiplication i.e. (ABC = A(BC):

`A=[[4       2        3],[1       1          2],[3         0          1]]`=`B=[[1        -1          1],[0         1            2],[2           -1          1]]` and  `C= [[1       2       -1],[3       0         1],[0         0         1]]` 


For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:

`A = [[1     -1],[0          2]] B=   [[-1       0],[2        1]]`and `C= [[0       1],[1     -1]]`


If 

 


Solve the matrix equations:

`[x1][[1,0],[-2,-3]][[x],[5]]=0`


Solve the matrix equations:

`[1  2   1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`


If `A=[[0,0],[4,0]]` find `A^16`


If `P=[[x,0,0],[0,y,0],[0,0,z]]` and `Q=[[a,0,0],[0,b,0],[0,0,c]]` prove that `PQ=[[xa,0,0],[0,yb,0],[0,0,zc]]=QP`


Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.

 

If A and B are square matrices of the same order, explain, why in general

(− B)2 ≠ A2 − 2AB + B2


Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that   (A + B)T = AT + BT


 For two matrices A and B,   \[A = \begin{bmatrix}2 & 1 & 3 \\ 4 & 1 & 0\end{bmatrix}, B = \begin{bmatrix}1 & - 1 \\ 0 & 2 \\ 5 & 0\end{bmatrix}\](AB)T = BT AT.

 


For the matrices A and B, verify that (AB)T = BT AT, where
\[A = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}\]

If the matrix \[A = \begin{bmatrix}5 & 2 & x \\ y & z & - 3 \\ 4 & t & - 7\end{bmatrix}\]  is a symmetric matrix, find xyz and t.
 

 


Express the matrix \[A = \begin{bmatrix}4 & 2 & - 1 \\ 3 & 5 & 7 \\ 1 & - 2 & 1\end{bmatrix}\] as the sum of a symmetric and a skew-symmetric matrix.

If  \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\]  , find AAT

 

If A = [aij] is a 2 × 2 matrix such that aij = i + 2j, write A.


If \[A = \begin{bmatrix}\cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\] is identity matrix, then write the value of α.


Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.


If `A=[[i,0],[0,i ]]` , n ∈ N, then A4n equals


If  \[\begin{bmatrix}\cos\frac{2\pi}{7} & - \sin\frac{2\pi}{7} \\ \sin\frac{2\pi}{7} & \cos\frac{2\pi}{7}\end{bmatrix}^k = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] then the least positive integral value of k is _____________.


If  \[A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\]then An (where n ∈ N) equals 

 


The matrix  \[A = \begin{bmatrix}0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0\end{bmatrix}\] is a


If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find A matrix Z such that X + Y + Z is a zero matrix


Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.


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