Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
Approach Solution - 1
Let A=\(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
We know that \(A = IA\)
⇒ \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\) = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}A\)
⇒ \(\begin{bmatrix} 1 & \frac32\\ 5 & 7 \end{bmatrix}\)=\(\begin{bmatrix} \frac12 & 0\\ 0 & 1 \end{bmatrix}\)\(A\) \((R_1 \rightarrow (\frac{1}{2}R_1) )\)
⇒ \(\begin{bmatrix} 1 & \frac32\\ 0 & \frac{-1}{2} \end{bmatrix}\)= \(\begin{bmatrix} \frac12 & 0\\ -\frac{5}{2} & 1 \end{bmatrix}A\) \( (R_2→ R_2-5R_1) \)
⇒ \(\begin{bmatrix} 1 & 0\\ 0 & \frac{-1}{2} \end{bmatrix}\)\(=\begin{bmatrix} -7 & 3\\ -\frac{5}{2} & 1 \end{bmatrix}A\) \((R_1→ R_1+3R_2)\)
⇒ \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\) = \(\begin{bmatrix} -7 & 3\\ 5 & -2 \end{bmatrix}A\) \((R_2\rightarrow-2R_1)\)
\(A^{-1}\)= \(\begin{bmatrix} -7 & 3\\ 5 & -2 \end{bmatrix}\)
Approach Solution -2
\(\mathbf{A} = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix}\)
we use the formula for the inverse of a \(2 \times 2\) matrix:
\(\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)
where \(\mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) and \(\det(\mathbf{A}) = ad - bc\).
First, we calculate the determinant \(\det(\mathbf{A})\):
\(\det(\mathbf{A}) = (2)(7) - (3)(5) = 14 - 15 = -1\)
Since the determinant is not zero, the inverse exists. Now, we apply the formula for the inverse:
\(\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{bmatrix} 7 & -3 \\ -5 & 2 \end{bmatrix}\)
Substituting \(\det(\mathbf{A}) = -1\):
\(\mathbf{A}^{-1} = \frac{1}{-1} \begin{bmatrix} 7 & -3 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} -7 & 3 \\ 5 & -2 \end{bmatrix}\)
So, the answer is \(\begin{bmatrix} -7 & 3 \\ 5 & -2 \end{bmatrix}\).
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
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. \(\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}\)
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
Concepts Used:
Invertible matrices
A matrix for which matrix inversion operation exists, given that it satisfies the requisite conditions is known as an invertible matrix. Any given square matrix A of order n × n is called invertible if and only if there exists, another n × n square matrix B such that, AB = BA = In, where In is an identity matrix of order n × n.
For example,
It can be observed that the determinant of the following matrices is non-zero.
