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Question: Let A be a \[2 \times 3\] matrix, whereas B be a \[3 \times 2\] matrix. If det (AB) \[ = 4\], then t...

Let A be a 2×32 \times 3 matrix, whereas B be a 3×23 \times 2 matrix. If det (AB) =4 = 4, then the value of det (BA) is
A.−4 - 4
B.22
C.−2 - 2
D.00

Explanation

Solution

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 2×32 \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 3×23 \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).

{{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.