Find the transpose of each of the following matrices:
I.\(\begin{bmatrix}5\\\frac{1}{2}\\-1\end {bmatrix}\)
II.\(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)
III.\(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)
Find the transpose of each of the following matrices:
I.\(\begin{bmatrix}5\\\frac{1}{2}\\-1\end {bmatrix}\)
II.\(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)
III.\(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)
Solution and Explanation
(i) Let A=\(\begin{bmatrix}5\\\frac{1}{2}\\-1\end {bmatrix}\)
then A-T= \(\begin{bmatrix}5&\frac{1}{2}&-1\end{bmatrix}\)
(ii)Let A= \(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)
then A-T= \(\begin{bmatrix}1&2\\-1&3\end{bmatrix}\)
(iii)Let A= \(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)
then A-T = \(\begin{bmatrix}-1&\sqrt3&2\\5&5&3\\6&6&-1\end{bmatrix}\)
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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 Matrices Questions
In the matrix A= \(\begin{bmatrix} 2 & 5 & 19&-7 \\ 35 & -2 & \frac{5}{2}&12 \\ \sqrt3 & 1 & -5&17 \end{bmatrix}\),write:
I. The order of the matrix
II. The number of elements
III. Write the elements a13, a21, a33, a24, a23If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Construct a 3×4 matrix, whose elements are given by
I. \(a_{ij}=\frac{1}{2}\mid -3i+j\mid\)
II. \(a_{ij}=2i-j\)Find the value of x, y, and z from the following equation:
I.\(\begin{bmatrix} 4&3&\\x&5\end{bmatrix}=\begin{bmatrix}y&z\\1&5\end{bmatrix}\)II. \(\begin{bmatrix}x+y&2\\5+z&xy\end{bmatrix}=\begin{bmatrix}6&2\\5&8\end{bmatrix}\)
III. \(\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}\)
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Concepts Used:
Transpose of a Matrix
The matrix acquired by interchanging the rows and columns of the parent matrix is called the Transpose matrix. The transpose matrix is also defined as - “A Matrix which is formed by transposing all the rows of a given matrix into columns and vice-versa.”
The transpose matrix of A is represented by A’. It can be better understood by the given example:
Now, in Matrix A, the number of rows was 4 and the number of columns was 3 but, on taking the transpose of A we acquired A’ having 3 rows and 4 columns. Consequently, the vertical Matrix gets converted into Horizontal Matrix.
Hence, we can say if the matrix before transposing was a vertical matrix, it will be transposed to a horizontal matrix and vice-versa.
Read More: Transpose of a Matrix