Question

For matrix A,
 $ \left| {0A} \right| = $
This question has multiple correct options.
A.0
B. $ \left| A \right| $
C. $ \left| 0 \right| $
D.None

Answer 1

Hint: A rectangular array or table of numbers, expressions, or symbols arranged into rows and columns is called a matrix. A matrix whose all the elements are zero is called a zero matrix and is represented by 0. Two matrices are multiplied with each other to give another matrix as a result. For multiplying two matrices we do the dot product of rows of the first matrix with the columns of the other matrix. So, by multiplying the zero matrix with matrix A, we can find out the correct answer.

Complete step-by-step answer:
Let A and 0 be square matrices of order 2.
Each element of a zero matrix is zero, so on multiplying a zero matrix with any other matrix, we get –
 $ 0A = \left( {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}} \\
  {{a_{21}}}&{{a_{22}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  {0 \times {a_{11}} + 0 \times {a_{21}}}&{0 \times {a_{12}} + 0 \times {a_{22}}} \\
  {0 \times {a_{11}} + 0 \times {a_{22}}}&{0 \times {a_{12}} + 0 \times {a_{22}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right) $
That is a zero matrix. So, $ \left| {0A} \right| = \left| 0 \right| $
Hence option (C) is the correct answer.
The determinant of a matrix is given as –
 $ \left| A \right| = \left| {\left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&{{a_{12}}} \\
  {{a_{21}}}&{{a_{22}}}
\end{array}} \right)} \right| = {a_{11}} \times {a_{22}} - {a_{12}} \times {a_{21}} $
So, the determinant of the zero matrix is $ \left| {\left( {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right)} \right| = 0 \times 0 - 0 \times 0 = 0 $
Hence, the determinant of a zero matrix is zero, so option (A) is also the correct answer.
So, the correct answer is “Option A and C”.

Note: A square matrix is a matrix in which the number of rows is equal to the number of columns. Every square matrix can be represented by a scalar value which can be calculated using the elements of the matrix and it encodes certain properties of the linear transformation described by the matrix. The determinant of any matrix A is represented as $ \left| A \right| $ .