Let $$A = {\left[ {{a_{ij}}} \right]_{m\, \times m}}$$ be a matrix and $$C = {\left[ {{c_{ij}}} \right]_{m\, \times m}}$$ be another matrix where $${c_{ij}}.$$ is the cofactor of $${a_{ij}}.$$ Then, what is the value of $$\left| {AC} \right|\,?$$
A.
$${\left| A \right|^{m - 1}}$$
B.
$${\left| A \right|^{m}}$$
C.
$${\left| A \right|^{m + 1}}$$
D.
Zero
Answer :
$${\left| A \right|^{m + 1}}$$
Solution :
Let $$A = {\left[ {{a_{ij}}} \right]_{m\, \times m}}$$ be a matrix and $$C = {\left[ {{c_{ij}}} \right]_{m\, \times m}}$$ be another matrix where $${c_{ij}}$$ is the cofactor of $${a_{ij}}.$$
∴ The value of $$\left| {AC} \right| = {\left| A \right|^{m + 1}}$$
Releted MCQ Question on Algebra >> Matrices and Determinants
Releted Question 1
Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then
A.
$$C$$ is empty
B.
$$B$$ has as many elements as $$C$$
C.
$$A = B \cup C$$
D.
$$B$$ has twice as many elements as elements as $$C$$
Let $$a, b, c$$ be the real numbers. Then following system of equations in $$x, y$$ and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$ has