Question
Download Solution PDFIf A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), then which of the following statements are correct?
A. A is a square matrix
B. A−1 exists
C. A is a symmetric matrix
D. |A| = 19
E. A is a null matrix
Choose the correct answer from the options given below.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFA. In the above matrix A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), we can see, the number of rows and columns are 2 respectively. Since the order of the matrix is 2 × 2, hence A is a square matrix.
B. The given 2 × 2 matrix \(A = \left[ {\begin{array}{*{20}{c}} 2&-3\\ 3&5 \end{array}} \right]\)
We first find the determinant of A.
Det A = (2 × 5) - (3 × -3) = 10 + 9 = 19
∴ |A| = 19
Since, |A| ≠ 0 ⇒ A-1 exists.
C. To know if a matrix is symmetric, find the transpose of that matrix. If the transpose of that matrix is equal to itself, it is a symmetric matrix. That is A = AT
Here A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\) then AT = \(\left[\begin{array}{cc}2 & 3 \\-3 & 5\end{array}\right]\)
Here, A ≠ AT
Thus A is not a symmetric matrix.
D. We have already derived |A| = 19.
E. Null Matrix: If in a matrix all the elements are zero then it is called a null matrix. It is also called a zero matrix. Here we can see A is not null matrix.
Thus A, B, D is the correct answer.
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