\((A^3)^{-1} = (A^{-1})^3\), where A is a square matrix and \(|A| \ne 0\).
True
\((aA)^{-1} = \dfrac{1}{a} A^{-1}\), where a is any real number and A is a square matrix.
False
\(|A^{-1}| = |A|^{-1}\), where A is a non-singular matrix.
False
If A and B are matrices of order 3 and \(|A| = 5\), \(|B| = 3\), then \(|3AB| = 27 \times 5 \times 3 = 405\).
True
If the value of a third order determinant is 12, then the value of the determinant formed by replacing each element by its cofactor will be 144.
True
\(|x+1\; x+2\; x+a;\; x+2\; x+3\; x+b;\; x+3\; x+4\; x+c| = 0\), where a, b, c are in A.P.
True
\(\operatorname{adj} A = |A| A^2\), where A is a square matrix of order two.
False
The determinant \(|\sin A\; \cos A\; \sin A + \cos B;\; \sin B\; \cos A\; \sin B + \cos B;\; \sin C\; \cos A\; \sin C + \cos B|\) is equal to zero.
True
If the determinant \(|x+a\; p+u\; l+f;\; y+b\; q+v\; m+g;\; z+c\; r+w\; n+h|\) splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.
True
Let \(|a\; p\; x;\; b\; q\; y;\; c\; r\; z| = 16\), then \(|p+x\; a+x\; a+p;\; q+y\; b+y\; b+q;\; r+z\; c+z\; c+r| = 32\).
True
The maximum value of the determinant \(|1\; 1\; 1;\; 1\; (1+\sin \theta)\; 1;\; 1\; 1\; 1+\cos \theta|\) is \(\dfrac{1}{2}\).
True