Question

Let A be a \[2 \times 3\] matrix, whereas B be a \[3 \times 2\] matrix. If det (AB) \[ = 4\], then the value of det (BA) is
A.\[ - 4\]
B.\[2\]
C.\[ - 2\]
D.\[0\]

Answer 1

Hint: In order to solve the question given above, you need to know about matrices and determinants. A matrix is a rectangular table of numbers or expressions which is arranged in rows and columns whereas determinant is a scalar value which is a function of the entries of a square matrix.

Complete step by step solution:
First, we have to create the matrix A and matrix B.
Since A is a \[2 \times 3\] matrix, let it be: \[A = \left[ {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{a_4}}&{{a_5}}&{{a_6}}
\end{array}} \right]\].
Also, B is a \[3 \times 2\] matrix, so, let it be: \[B = \left[ {\begin{array}{*{20}{c}}
  {{b_1}}&{{b_2}} \\
  {{b_3}}&{{b_4}} \\
  {{b_5}}&{{b_6}}
\end{array}} \right]\].
Now we have to find (BA)
For this, we multiply both the matrices.
Now, we have to find the determinant of (BA).
\[\left( {BA} \right) = \left| {\left. {\begin{array}{*{20}{c}}
  {{b_1}{a_1} + {b_2}{a_4}}&{{b_1}{a_2} + {b_2}{a_5}}&{{b_1}{a_3} + {b_2}{a_6}} \\
  {{b_3}{a_1} + {b_4}{a_4}}&{{b_3}{a_2} + {b_4}{a_5}}&{{b_3}{a_3} + {b_4}{a_6}} \\
  {{b_5}{a_1} + {b_6}{a_4}}&{{b_5}{a_2} + {b_6}{a_5}}&{{b_5}{a_3} + {b_6}{a_6}}
\end{array}} \right|} \right.\].
So, the determinant is:
\[\left( {BA} \right) = \left| {\left. {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\
  {{a_4}}&{{a_4}}&{{a_6}} \\
  0&0&0
\end{array}} \right|} \right. \times \left| {\left. {\begin{array}{*{20}{c}}
  {{b_1}}&{{b_2}}&0 \\
  {{b_3}}&{{b_4}}&0 \\
  {{b_5}}&{{b_6}}&0
\end{array}} \right|} \right.\],
Therefore, from the above calculations, we get that \[\left( {BA} \right) = 0\].
Hence, the correct option is d) \[0\].
So, the correct answer is “Option D”.

Note: While solving questions similar to the one given above you need to keep few concepts in your mind. 1) A rectangular table of symbols or numbers arranged in rows and columns is called a matrix. Plural form is called matrices. 2) In the dimensions of a matrix \[p \times q\], \[p\] is the number of rows whereas \[q\] is the number of columns of a matrix 3) determinant refers to a scalar value that is a function of the entries of a square matrix. Its input is a square matrix and its output is a number.