Advertisements
Advertisements
प्रश्न
Assertion: Let the matrices A = `((-3, 2),(-5, 4))` and B = `((4, -2),(5, -3))` be such that A100B = BA100
Reason: AB = BA implies AB = BA for all positive integers n.
पर्याय
Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
Assertion is true and Reason is false.
Assertion is false and Reason is true.
Advertisements
उत्तर
Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
Explanation:
We have, A = `[(-3, 2),(-5, 4)]`, B = `[(4, -2),(5, -3)]`
Now, AB = `[(-3, 2),(-5, 4)][(4, -2),(5, -3)] = [(-2, 0),(0, -2)]`
And BA = `[(4, -2),(5, -3)][(-3, 2),(-5, 4)] = [(-2, 0),(0, -2)]`
Hence, AB = BA
Now, A2 = `[(-3, 2),(-5, 4)][(-3, 2),(-5, 4)] = [(-1, 2),(-5, 6)]`
So, A2B = `[(-1, 2),(-5, 6)][(4, -2),(5, -3)] = [(6, -4),(10, -8)]`
And BA2 = `[(4, -2),(5, -3)][(-1, 2),(-5, 6)] = [(6, -4),(10, -8)]`
Hence, A2B = BA2
If, AB = BA and A2B = BA2...............
Therefore, AnB = BAn
Also, A100B = BA100
Hence, Assertion and Reason both are true.
APPEARS IN
संबंधित प्रश्न
Find the value of a, b, c and d from the equation:
`[(a - b, 2a + c),(2a - b, 3c + d)] = [(-1, 5),(0, 13)]`
`A = [a_(ij)]_(m xx n)` is a square matrix, if ______.
if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`
Find the matrix X so that X`[(1, 2, 3),(4, 5, 6)]= [(-7, -8, -9),(2, 4, 6)]`
If A is a square matrix such that A2 = A, then (I + A)3 – 7A is equal to ______.
Determine the product `[(-4,4,4),(-7,1,3),(5,-3,-1)][(1,-1,1),(1,-2,-2),(2,1,3)]` and use it to solve the system of equations x - y + z = 4, x- 2y- 2z = 9, 2x + y + 3z = 1.
Show that a matrix A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]` is unitary.
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[9 sqrt(2) -3]`
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(6, 0),(0, 6)]`
Identify the following matrix is singular or non-singular?
`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
Construct the matrix A = [aij]3 × 3 where aij = i − j. State whether A is symmetric or skew-symmetric.
If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, Show that A2 – 4A is a scalar matrix
Answer the following question:
If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find B + C – A
Answer the following question:
If A = `[(1, 2, 3),(2, 4, 6),(1, 2, 3)]`, B = `[(1, -1, 1),(-3, 2, -1),(-2, 1, 0)]`, show that AB and BA are both singular matrices
If A = `[(6, 0),("p", "q")]` is a scalar matrix, then the values of p and q are ______ respectively.
If A = `[(2, 0, 0),(0, 1, 0),(0, 0, 1)]`, then |adj (A)| = ______
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
2X + 3Y = `[(2, 3),(4, 0)]`, 3Y + 2Y = `[(-2, 2),(1, -5)]`
If A = `[(0,0,0),(0,0,0),(0,1,0)]` then A is ____________.
If the matrix A `= [(5,2,"x"),("y",2,-3),(4, "t",-7)]` is a symmetric matrix, then find the value of x, y and t respectively.
The matrix `[(0,5,-7),(-5,0,11),(7,-11,0)]` is ____________.
If A is a square matrix such that A2 = A, then (I + A)2 - 3A is ____________.
A matrix is said to be a row matrix, if it has
If all the elements are zero, then matrix is said to be
A = `[a_(ij)]_(m xx n)` is a square matrix, if
Find X, If `[X - 5 - 1] [(1, 0, 2),(0, 2, 1),(2, 0, 3)][(x),(4),(1)] ` = 0
A diagonal matrix in which all diagonal elements are same, is called a ______ matrix.
Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠ 0. Consider the following two statements:
(P) If A1I2, then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,
where I2 denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then ______.
If `[(1, 2, 1),(2, 3, 1),(3, a, 1)]` is non-singular matrix and a ∈ A, then the set A is ______.
Matrices are classified into different types based on which two features?
A matrix with only one column is called a:
Which general form represents a row matrix?
If \[m=n\], what type and order does the matrix have?
Which condition holds for a diagonal matrix when \[i\ne j\]?
A scalar matrix is a diagonal matrix in which:
An identity matrix (unit matrix) is a square matrix with:
What happens when any matrix is multiplied by an identity matrix?
Which matrix is an identity matrix?
What is the notation for a zero matrix?
