Question

Let A be \[3 \times 3\] matrix and B be an adjoint matrix. If \[|B| = 64\] , then find \[|A|\]

Answer 1

Hint: Here the question is related to the topic matrix. The matrix is a square matrix and they have given an adjoint matrix and determinant value of the adjoint matrix. By using this we have to determine the value of the determinant of the matrix A. By using the properties of the determinant of the adjoint matrix we determine the value of the determinant of matrix A.

Complete step-by-step answer:
In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that particular matrix. The determinant is a scalar value that is a function of the entries of a square matrix.
Now consider the given question and we have
\[ \Rightarrow B = adj(A)\]
Take the determinant on both sides of the above inequality
\[ \Rightarrow \left| B \right| = \left| {adj(A)} \right|\]
As we know that \[\left| {adj(A)} \right| = {\left| A \right|^2}\], therefore we have
\[ \Rightarrow \left| B \right| = {\left| A \right|^2}\]
By the question we know the value of determinant B and it is given as \[|B| = 64\]. On substituting we have
\[ \Rightarrow 64 = {\left| A \right|^2}\]
Taking square root on the both sides we have
\[ \Rightarrow \sqrt {64} = \sqrt {{{\left| A \right|}^2}} \]
In RHS the square and square root will get cancels and the square root of 64 is plus or minus 8 so we have
\[ \Rightarrow |A| = \pm 8\]
Therefore the value of \[|A| = \pm 8\]
Hence we have determined the value.
So, the correct answer is “\[|A| = \pm 8\]”.

Note: The main concept we have to remember is \[\det (adj(A)) = \det {(A)^{n - 1}}\] For this question the value of n is 3, since it is \[3 \times 3\] matrix. Only this formula helps us to make the problem much easier. Then we have to mention about the signs that is plus and minus because both the values that is plus term and minus term answer will be same