# Symmetric Matrix

## Trending Questions

**Q.**Let A and B be 3×3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2−B2A2)X=O, where X is a 3×1 column matrix of unknown variables and O is a 3×1 null matrix, has

- a unique solution
- exactly two solutions
- infinitely many solutions
- no solution

**Q.**

Let ${\left(2{x}^{2}+3x+4\right)}^{10}=\underset{r=0}{\overset{20}{\xe2\u02c6\u2018}}{a}_{r}{x}^{r}$, then $\frac{{a}_{7}}{{a}_{13}}=$

**Q.**If A is a symmetric matrix and B is a skew- symmetrix matrix such that A+B=[235−1], then AB is equal to :

- [−4214]
- [4−21−4]
- [4−2−1−4]
- [−4−2−14]

**Q.**

If A is a skew symmetric matrix, then A2 is a

**Q.**If A is a skew-symmetric matrix of order 3, then the matrix A4 is

- symmetric
- skew symmetric
- diagonal
- none of those

**Q.**

Express the matrix B=⎡⎢⎣2−2−4−1341−2−3⎤⎥⎦ as the sum of a symmetric and a skew symmetric matrix.

**Q.**Let A=⎛⎜⎝1−1001−1001⎞⎟⎠ and B=7A20−20A7+2I, where I is an identity matrix of order 3×3. If B=[bij], then b13 is equal to

**Q.**If A is a square matrix, then

- AAT is symmetric matrix and ATA is skew-symmetric matrix.
- AAT is skew-symmetric matrix and ATA is symmetric matrix.
- Both AAT and ATA are symmetric matrices.
- Both AAT and ATA are skew-symmetric matrices.

**Q.**Let A be the set of all 3×3 symmetric matrices, all of whose entries are either 0 or 1, five of these are 1 and four of them are 0

Consider the system of linear equations A⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣100⎤⎥⎦, then which of the following is/are correct?

- The total number of possible matrices in A is 12.
- The number of matrices A in set A for which the system of linear equations has a unique solution is 6.
- The number of matrices A in set A for which the system of linear equations has a unique solution is 4.
- The number of matrices A in set A for which the system of linear equations is inconsistent is more than 2.

**Q.**Which among the following graph(s) is/are even function?

**Q.**

Show that the matrix B' AB is symmetric or skew-symmetric according to A which is symmetric or skew -symmetric.

**Q.**

A square matrix A is said to be a symmetric matrix if

A=−AT

A=AT

A=¯A

A=−¯A

**Q.**If A=⎡⎢⎣52ab0−34c−7⎤⎥⎦ is a symmetric matrix, then find the value of (a+b+c).

- 3
- 11
- 15
- 2

**Q.**If the matrix⎡⎢⎣0a32b−1c10⎤⎥⎦is a skew symmetric matrix, then the value of |a+b+c| is

**Q.**

Is a zero matrix symmetric?

**Q.**If A is a square matrix, then which of the following is correct ?

(a) AAT is symmetric matrix and ATA is skew-symmetric matrix.

(b) AAT is skew-symmetric matrix and ATA is symmetric matrix.

(c) Both AAT and ATA are symmetric matrices.

(d) Both AAT and ATA are skew-symmetric matrices.

**Q.**If A , B are symmetric matrices of same order, then AB − BA is a A. Skew symmetric matrix B. Symmetric matrix C. Zero matrix D. Identity matrix

**Q.**Let ABC=I then tr(ABC+BCA+CAB) is

(where order of matrices A, B, C is 3 and tr(A) is sum of the principle diagonal elements in A)

**Q.**If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.

**Q.**

What Are The Eigenvalues Of A Symmetric Matrix?

**Q.**For the matrix A=[1567], verify that

(i) (A+A′) is a symmetric matrix.

(ii) (A−A′) is a skew symmetric matrix.

**Q.**

In a skew-symmetric matrix, the diagonal elements are all.

One.

Zero.

Different from each other.

Non-zero.

**Q.**If A and B are two non singular matrices and both are symmetric and commute each other then

- A−1B is symmetric but A−1B−1 is not symmetric
- Both A−1B and A−1B−1 are symmetric
- Neither A−1B nor A−1B−1 are symmetric
- A−1B−1 is symmetric but A−1B is not symmetric

**Q.**

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

⎡⎢⎣6−22−23−12−13⎤⎥⎦

**Q.**Distinct prime numbers p, q, r satisfy the equation 2pqr+50pq=7pqr+55pr=8pqr+12qr=A, for some positive integer A. Then A is

- 660
- 1980
- 388
- 180

**Q.**

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

[15−12]

**Q.**If A and B are non-singular matrix of order 3 and (I−AB) is invertible, then which of the following statements is/are correct:

- I−BA is not invertible
- I−BA is invertible
- inverse of (I−BA) is I+B(I−AB)−1A
- inverse of (I−BA) is I+A(I−BA)−1B

**Q.**

Two matrices are multiplied by multiplying their corresponding elements.

False

True

**Q.**Show that the matrix is symmetric or skew symmetric according as A is symmetric or skew symmetric.

**Q.**Let P be a non-singular matrix and I+P+P2+⋯+Pn=O. Then P−1 is equal to (where n∈Z+)

- Pn
- P
- Pn−1
- I