Question:
If
\[
A=
\begin{bmatrix}
3 & -3 & 4 \\
2 & -3 & 4 \\
0 & -1 & 1
\end{bmatrix},
\]
then \(AA^T\) is a
If
\[
A=
\begin{bmatrix}
3 & -3 & 4 \\
2 & -3 & 4 \\
0 & -1 & 1
\end{bmatrix},
\]
then \(AA^T\) is a
Show Hint
For any matrix \(A\), the matrix
\[
AA^T
\]
is always symmetric because
\[
(AA^T)^T=AA^T.
\]
Updated On: Jun 22, 2026
- Symmetric matrix
- Skew-Symmetric matrix
- Singular matrix
- Inverse of \(A\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understand the given expression.
We are given a matrix \[ A= \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix} \] We need to identify the nature of \[ AA^T \] where \(A^T\) represents the transpose of matrix \(A\).
Step 2: Use the transpose property.
For any matrix \(A\), consider \[ AA^T \] Now take transpose on both sides: \[ (AA^T)^T \] Using the property \[ (AB)^T=B^TA^T, \] we get \[ (AA^T)^T=(A^T)^T A^T \] Since \[ (A^T)^T=A, \] therefore, \[ (AA^T)^T=AA^T \]
Step 3: Apply the definition of symmetric matrix.
A matrix \(M\) is called symmetric if \[ M^T=M \] Here, if \[ M=AA^T, \] then \[ M^T=(AA^T)^T=AA^T=M \] So, \[ AA^T \] is a symmetric matrix.
Step 4: Check why other options are not correct.
A skew-symmetric matrix satisfies \[ M^T=-M \] but here \[ (AA^T)^T=AA^T \] so it is not skew-symmetric.
A singular matrix must have determinant zero, but the question asks the general nature of \(AA^T\), and the standard property is that \(AA^T\) is symmetric.
Also, \[ AA^T \] cannot be called the inverse of \(A\) unless it satisfies \[ AA^T=I, \] which is not given here.
Step 5: Final conclusion.
Thus, \[ AA^T \] is a symmetric matrix.
Therefore, \[ \boxed{\text{Symmetric matrix}} \]
We are given a matrix \[ A= \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix} \] We need to identify the nature of \[ AA^T \] where \(A^T\) represents the transpose of matrix \(A\).
Step 2: Use the transpose property.
For any matrix \(A\), consider \[ AA^T \] Now take transpose on both sides: \[ (AA^T)^T \] Using the property \[ (AB)^T=B^TA^T, \] we get \[ (AA^T)^T=(A^T)^T A^T \] Since \[ (A^T)^T=A, \] therefore, \[ (AA^T)^T=AA^T \]
Step 3: Apply the definition of symmetric matrix.
A matrix \(M\) is called symmetric if \[ M^T=M \] Here, if \[ M=AA^T, \] then \[ M^T=(AA^T)^T=AA^T=M \] So, \[ AA^T \] is a symmetric matrix.
Step 4: Check why other options are not correct.
A skew-symmetric matrix satisfies \[ M^T=-M \] but here \[ (AA^T)^T=AA^T \] so it is not skew-symmetric.
A singular matrix must have determinant zero, but the question asks the general nature of \(AA^T\), and the standard property is that \(AA^T\) is symmetric.
Also, \[ AA^T \] cannot be called the inverse of \(A\) unless it satisfies \[ AA^T=I, \] which is not given here.
Step 5: Final conclusion.
Thus, \[ AA^T \] is a symmetric matrix.
Therefore, \[ \boxed{\text{Symmetric matrix}} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Matrices Questions
- If there exists a \( k^{th} \) order non-singular sub-matrix in a matrix \( P \) of order \( m \times n \), then the rank \( (\rho) \) of \( P \)
- If \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \), then which one of the following is True?
- In a matrix \( A \), if all the sub matrices of \( k^{th} \) order are singular and there is one non-singular sub matrix of order \( r \) \( (r<k) \), then the rank \( (\rho) \) of the matrix \( A \)
- The number of ordered pairs $ (x, y) $ for which $ A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 2 & x \\ y & 1 & 2 \end{pmatrix} $ is a singular and symmetric matrix is:
Let $ A = \begin{bmatrix} 2 & 1 & 3 & -1 \\1 & -2 & 2 & -3 \end{bmatrix}, B = \begin{bmatrix} 2 & 1 & 0 & 3 \\1 & -1 & 2 & 3 \end{bmatrix} $, and the equation $ 2A + 3B - 5C = 0 $. Find the matrix $ C $.
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions