Consider the determinant
Δ = $\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33}\end{array}\right|$
If a13 = yz, a23 = zx, a33 = xy and the minors of a13, a23, a33 are respectively (z − y), (z − x), (y − x) then what is the value of Δ ?
If A = $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & −\cos \theta\end{array}\right)$, then which of the following are correct?
1. A + adj A is a null matrix
2. A−1 + adj A is a null matrix
3. A − A−1 is a null matrix
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If X is a matrix of order 3 × 3, Y is a matrix of order 2 × 3 and Z is a matrix of order 3 × 2, then which of the following are correct?
1. (ZY)X is a square matrix having 9 entries.
2. Y(XZ) is a square matrix having 4 entries.
3. X(YZ) is not defined.
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