Question

Choose the correct statement.\[\]
A. Determinant is a square matrix. \[\]
B. Determinant is a number associated with matrix.\[\]
C. Determinant is a number associated with a square matrix.
D .None of these\[\]

Answer 1

Hint: We recall the definitions of the matrix, square matrix, and determinant to check whether a determinant is a matrix or whether we can find the fins the determinant of any matrix or just square matrix. We choose the correct option accordingly. \[\]

Complete step by step answer:
We know that matrix is a rectangular array with homogeneous entries. If the number of rows is denoted as $m$ and the number of columns is denoted as $n$ then the order of the matrix says $A$ is given by $m\times n.$ The square matrix is a matrix whose number of rows and columns are equal which means $m=n$.If the number of rows or columns of a square matrix is $n$ then we say the square matrix $A$ is of order $n.$ \[\]
The determinant is a scalar value computed from a square matrix in its first row or first column. The determinant of square matrix is denoted as $\det \left( A \right)$ or $\left| A \right|$. We illustrate it with the standard square matrix $A$ of order 2 the homogeneous entries. We have
\[A=\left[ \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right]\]
Then determinant value is obtained by cross multiplying and subtracting. We have
\[A=\left| \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right|=ad-bc\]

The determinant of a square matrix $A$ of order 3 is given by
\[A=\left[ \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right]\Rightarrow A=\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right|\]
We compute it by expansion of the first row taking an alternative sign on $a,b,c$ and multiplying the left out $2\times 2$ determinant which does not contain the $a,b,c$ respectively.
\[\begin{align}
  & A=\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right| \\
 & =a\left| \begin{matrix}
   e & f \\
   h & i \\
\end{matrix} \right|-b\left| \begin{matrix}
   d & f \\
   g & i \\
\end{matrix} \right|+c\left| \begin{matrix}
   d & e \\
   g & h \\
\end{matrix} \right| \\
\end{align}\]
So we always need a square matrix to find determinant. We now check the options. We have in option A, Determinant is a square matrix which is false because determinant is a scalar and matrix is array. We see in B, Determinant is a number associated with matrix which is false as we can only find the determinant of square matrix which we see written correctly in option C. \[\]

Note:
 We note that the determinant of a square matrix $A$ and its transpose ${{A}^{T}}$ are equal which means $\det \left( A \right)=\det \left( {{A}^{T}} \right)$. We can also find the determinant of inverse of a square matrix $A$ using the formula $\det \left( {{A}^{-1}} \right)=\dfrac{1}{\det \left( A \right)}.$