Let \(A\), \(B\) and \(C\) be square matrices of order 2 such that \(AB = AC\). Which of the statements given below is/are correct? I. \(B\) and \(C\) are not necessarily equal if \(A\) is a singular matrix. II. \(B\) and \(C\) are equal if \(A\) is a non-singular matrix. Select the answer using the code given below.
Both I and II
If \(A\) is singular, \(A^{-1}\) does not exist, so \(AB=AC\) does not force \(B=C\) — statement I is correct. If \(A\) is non-singular, multiplying both sides by \(A^{-1}\) gives \(B=C\) — statement II is correct. Hence both statements hold.
Similar Questions
What should be the value of x so that the matrix \(\left( {\begin{array}{*{20}{c}} 2&4\\ { - 8}&{\rm{x}} \end{array}} \right)\) does not have an inverse?
What is the adjoint of the matrix \(\left( {\begin{array}{*{20}{c}} {\cos \left( { - \theta } \right)}&{ - \sin \left( { - \theta } \right)}\\ { - \sin \left( { - \theta } \right)}&{\cos \left( { - \theta } \right)} \end{array}} \right)\) ?
What is the inverse of the matrix?
\(A = \left( {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta }&0\\ { - \sin \theta }&{\cos \theta }&0\\ 0&0&1 \end{array}} \right)\)
If A and B are two invertible square matrices of same order, then what is (AB) -1 equal to?
If \(B = \left[ {\begin{array}{*{20}{c}} 3&2&0\\ 2&4&0\\ 1&1&0 \end{array}} \right]\) , then what is adjoint of B equal to?
For a square matrix A, which of the following properties hold?
1) (A -1 )-1 = A
2) \(\det \left( {{A^{ - 1}}} \right) = \frac{1}{{detA}}\)
3) (λA) -1 = λA -1 where λ is a scalar
Select the correct answer using the code given below:If A is a square matrix, then what is adj (A -1 ) – (adj A) -1 equal to?
The matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&3&2\\ 1&{{\rm{x}} - 1}&1\\ 2&7&{{\rm{x}} - 3} \end{array}} \right]\)
Will have inverse for every real number x except forIf $A = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix}$, then what is $A + 3A^{-1}$ equal to, where $A$ is a matrix of order 2?
If \({\rm{A}} = \left[ {\begin{array}{c} 1&{ - 1}\\ 2&3 \end{array}} \right]{\rm{\;\;and\;B}} = \left[ {\begin{array}{c} 2&3\\ { - 1}&{ - 2} \end{array}} \right]\) , then which of the following is/are correct?
1. AB (A -1 B-1 ) is a unit matrix
2. (AB) -1 = A -1 B-1
Select the correct answer using the code given below:
Important Questions from Adjoint and Inverse of a Square Matrix
What should be the value of x so that the matrix \(\left( {\begin{array}{*{20}{c}} 2&4\\ { - 8}&{\rm{x}} \end{array}} \right)\) does not have an inverse?
What is the adjoint of the matrix \(\left( {\begin{array}{*{20}{c}} {\cos \left( { - \theta } \right)}&{ - \sin \left( { - \theta } \right)}\\ { - \sin \left( { - \theta } \right)}&{\cos \left( { - \theta } \right)} \end{array}} \right)\) ?
The inverse of the matrix A = \(\left( {\begin{array}{} 1&1&3\\ 1&3&{ - 3}\\ { - 2}&{ - 4}&{ - 4} \end{array}} \right)\) is:
What is the inverse of the matrix?
\(A = \left( {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta }&0\\ { - \sin \theta }&{\cos \theta }&0\\ 0&0&1 \end{array}} \right)\)
If A and B are two invertible square matrices of same order, then what is (AB) -1 equal to?
