Advertisements
Advertisements
Question
If li, mi, ni, i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I, where \[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]
Advertisements
Solution
Given : \[\left( l_1 , m_1 , n_1 \right), \left( l_2 , m_2 , n_2 \right), \left( l_3 , m_3 , n_3 \right)\]are the direction cosines of three mutually perpendicular vectors in space.
\[\left. \begin{array}l_1^2 + m_1^2 + n_1^2 = 1 \\ l_2^2 + m_2^2 + n_2^2 = 1 \\ l_3^2 + m_3^2 + n_3^2 = 1\end{array} \right\} . . . . . \left( i \right)\]
\[\left. \begin{array}l_1 l_2 + m_1 m_2 + n_1 n_2 = 0 \\ l_2 l_3 + m_2 m_3 + n_2 n_3 = 0 \\ l_3 l_1 + m_3 m_1 + n_3 n_1 = 0\end{array} \right\} . . . . . \left( ii \right)\]
Let \[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]
\[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]
\[A A^T = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\begin{bmatrix}l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \\ n_1 & n_2 & n_3\end{bmatrix}\]
\[ \Rightarrow A A^T = \begin{bmatrix}{l_1}^2 + {m_1}^2 + {n_1}^2 & l_1 l_2 + m_1 m_2 + n_1 n_2 & l_3 l_1 + m_3 m_1 + n_3 n_1 \\ l_1 l_2 + m_1 m_2 + n_1 n_2 & {l_2}^2 + {m_2}^2 + {n_2}^2 & l_2 l_3 + m_2 m_3 + n_2 n_3 \\ l_3 l_1 + m_3 m_1 + n_3 n_1 & l_2 l_3 + m_2 m_3 + n_2 n_3 & {l_3}^2 + {m_3}^2 + {n_3}^2\end{bmatrix}\]
\[\]
From (i) and (ii), we get
\[A A^T = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = I\]
Hence proved.
APPEARS IN
RELATED QUESTIONS
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)]`
if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where n is any positive integer
Given `A = [(2,-3),(-4,7)]` compute `A^(-1)` and show that `2A^(-1) = 9I - A`
Find the non-singular matrices P & Q such that PAQ is in normal form where`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]`
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`
If A is a square matrix of order 3 with |A| = 4 , then the write the value of |-2A| .
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)]`
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)]`
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:
`[(10, -15, 27),(-15, 0, sqrt(34)),(27, sqrt(34), 5/3)]`
Identify the following matrix is singular or non-singular?
`[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]`
Identify the following matrix is singular or non-singular?
`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, Find (AT)T.
Find x, y, z If `[(0, -5"i", x),(y, 0, z),(3/2, -sqrt(2), 0)]` is a skew symmetric matrix.
Select the correct option from the given alternatives:
If A and B are square matrices of equal order, then which one is correct among the following?
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
Choose the correct alternative:
If A = `[(2, 0),(0, 2)]`, then A2 – 3I = ______
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)| = ______
If A and B are matrices of same order, then (3A –2B)′ is equal to______.
Show by an example that for A ≠ O, B ≠ O, AB = O
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","b"),("c", "-a")]`is a square root of the 2 x 2 identity matrix, then a, b, c satisfy the relation ____________.
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.
If a matrix A is both symmetric and skew-symmetric, then ____________.
The matrix `[(0,-5,8),(5,0,12),(-8,-12,0)]` is a ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
A matrix is said to be a row matrix, if it has
A = `[a_(ij)]_(m xx n)` is a square matrix, if
If 'A' is square matrix, such that A2 = A, then (7 + A)3 = 7A is equal to
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 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 ______.
If `A = [(1,-1,2),(0,-1,3)], B = [(-2,1),(3,-1),(0,2)],` then AB is a singular matrix.
