Advertisements
Advertisements
Question
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix} \]
\[ A^T = \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}\]
\[Now, \]
\[\left( A - A^T \right) = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix} - \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = \begin{bmatrix}2 - 2 & 3 - 4 \\ 4 - 3 & 5 - 5\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix} . . . \left( 1 \right)\]
\[ \left( A - A^T \right)^T = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix}^T \]
\[ \Rightarrow \left( A - A^T \right)^T = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right)^T = - \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = - \left( A - A^T \right)^T \left[ \text{Using eq} . \left( 1 \right) \right]\]
\[Thus, \left( A - A^T \right) \text{is a skew - symmetric matrix} .\]
APPEARS IN
RELATED QUESTIONS
If A is a square matrix, such that A2=A, then write the value of 7A−(I+A)3, where I is an identity matrix.
Find the value of x, y, and z from the following equation:
`[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]`
`A = [a_(ij)]_(mxxn)` is a square matrix, if ______.
if `A = [(0, -tan alpha/2), (tan alpha/2, 0)]` and I is the identity matrix of order 2, show that I + A = `(I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`
Given two matrices A and B
`A = [(1,-2,3),(1,4,1),(1,-3, 2)] and B = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`
find AB and use this result to solve the following system of equations:
x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1
A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?
If li, mi, ni, i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I, where \[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]
if `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`
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:
`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`
Identify the following matrix is singular or non-singular?
`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]`
Answer the following question:
If A = `[(1, 2),(3, 2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, -3)]`, show that AB is singular.
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.
State whether the following statement is True or False:
If A and B are two square matrices such that AB = BA, then (A – B)2 = A2 – 2AB + B2
If A = `[(2, 0, 0),(0, 1, 0),(0, 0, 1)]`, then |adj (A)| = ______
The matrix A = `[(0, 0, 5),(0, 5, 0),(5, 0, 0)]` is a ______.
If two matrices A and B are of the same order, then 2A + B = B + 2A.
AB = AC ⇒ B = C for any three matrices of same order.
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 ____________.
A square matrix A = [aij]nxn is called a diagonal matrix if aij = 0 for ____________.
If a matrix A is both symmetric and skew-symmetric, then ____________.
If `[(1,2),(3,4)],` then A2 - 5A is equal to ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
A square matrix B = [bÿ] m × m is said to be a diagonal matrix if all diagonal elements are
A diagonal matrix is said to be a scalar matrix if its diagonal elements are
If all the elements are zero, then matrix is said to be
The number of all possible matrices of order 3/3, with each entry 0 or 1 is
If the sides a, b, c of ΔABC satisfy the equation 4x3 – 24x2 + 47x – 30 = 0 and `|(a^2, (s - a)^2, (s - a)^2),((s - b)^2, b^2, (s - b)^2),((s - c)^2, (s - c)^2, c^2)| = p^2/q` where p and q are co-prime and s is semiperimeter of ΔABC, then the value of (p – q) is ______.
The minimum number of zeros in an upper triangular matrix will be ______.
Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the systems of linear equations (A2B2 – B2A2)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has ______.
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.
