Scalar Matrix

Q.

What is a scalar matrix?

Q. If A is orthogonal matrix of order 2 then det(adj 2A) =

(a) 4 (b) 16. (c) 27 (d) 64

Q. Let $A$ and $B$ be two square matrices of order $3$ such that $det\left(A\right)=5$ and $det\left(B\right)=2.$ Then the value of $det(det(B)\cdot A)$ is

1. $40$
2. $20$
3. $8$
4. $10$

Q. Suppose that a, b, c are all real numbers such that $a+b+c=1$. If the matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ is an orthogonal matrix, then which of the following is correct?

1. A must be a singular matrix.
2. $|A|=1$
3. $a^2+b^3+c^3-3abc=1$
4. atleast one of a, b, c must be zero.

Q. 2. Let A and B be real matrices of the form (shown in picture). Statement 1:AB-BA is always an invertible matrix. Statement 2: AB-BA is never an identity matrix.

Q. If determinant of matrix $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$ is A , then determinant of $\left|\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right|$ is

1. A
2. -A
3. $A^2$
4. -$A^2$

Q. Find $X$ and $Y$, if $X+Y=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$

Q. If A is a square matrix of order 3 such that $A^3=Iand(A+I{)}^3+(A-I{)}^3-6A=B$ where I is identity matrix of order 3, then

1. |B| = 3
2. |B| = 2
3. |B| = 4
4. |B| = 8

Q. Define a scalar matrix.

Q.

What do you mean by generalized eigenvector?

Q. Let $A$ be a $n^{th}$ order square matrix and matrix $B$ be its adjoint, then $|AB+K{I}_n|$ is equal to (where $K$ is a scalar quantity)

1. $(|A|+K{)}^{n-2}$
2. $(|A|+K{)}^n$
3. $(|A|+K{)}^{n-1}$
4. $(|A|+K{)}^{-1}$

Q. A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.

Q.

What are matrices in Math?

Q. If A = $\left|\begin{array}{ccc}ka& kb& kc\\ kd& ke& kf\\ kg& kh& ki\end{array}\right|$ and B = $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|$ them A/B =

1. 1
2. k
3. $k^3$
4. $k^9$

Q. If a square matrix $A$ is such that $det\left(A\right)=5$ and $det\left(2A\right)=320$, then the order of $A$ is

Q.

Which among the following is a scalar matrix?

1. 

2. 

3. 

4. 

Q. The number of integers $x$ satisfying
$-3x^4+\text{det}\left[\begin{array}{ccc}1& x& x^2\\ 1& x^2& x^4\\ 1& x^3& x^6\end{array}\right]=0$
is equal to

1. $2$
2. $1$
3. $5$
4. $8$

Q. If I is the identity matrix and A is a square matrix such that A² = A, then what is the value of (I + A)² = 3A?

Q. Let $A$ be a matrix such that $A\cdot \left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]$ is a scalar matrix and $|3A|=108$. Then $A^2$ equals :

1. $\left[\begin{array}{cc}36& 0\\ -32& 4\end{array}\right]$
2. $\left[\begin{array}{cc}4& 0\\ -32& 36\end{array}\right]$
3. $\left[\begin{array}{cc}4& -32\\ 0& 36\end{array}\right]$
4. $\left[\begin{array}{cc}36& -32\\ 0& 4\end{array}\right]$

Q.

If k is scalar $(k\ne 0)$, I is an identity matrix and A is a square matrix . Then which among the following will always be a scalar matrix.

1. $kA$

2. $kI$

3. $\frac{1}{k}.A$

4. $\frac{1}{k}.I$

Q. Solve for $x$ and $y$
$ax+by=c$
$bx+ay=1+c$

Q.

If f(x) = x^2 -5x +7 and let A be a matrix . f(A) is written as A^2 -5A +7I where I is an Identity matrix .Why 7 is written as 7I as It is a scalar and multiplying by matrix makes it a matrix .

Q. 64. If A, B, C are 3 x 3 matrix and det (A)=2, det (B)=4 and det (C)=1/2 . then the value ofdet (A^4B^{-1}) + det (adj(2C)) is equal to

Q. Let $I$ be an identity matrix of order $2\times 2$ and $P=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right].$ Then the value of $n\in N$ for which $P^n=5I-8P$ is equal to

Q.

If, then show that

Q. If $A$ is any square matrix of order $3\times 3$ then $\left|3A\right|$ is equal to

1. $3\left|A\right|$
2. $\frac{1}{3}\left|A\right|$
3. $27\left|A\right|$
4. $9\left|A\right|$

Q. If $A$ is a square matrix of order $3$, then $|(A-A^T{)}^{2019}|=$

1. $1$
2. $2019$
3. $0$
4. $\text{None of these}$

Q. Let $A$ and $B$ are square matrices of order $2,$ such that $|A|=6$ and $|B|=5.$ If the value of $|3A||4B|=\alpha ,$ then the value of $\frac{\alpha }{144}$ is

Q.

Find the determinant value of an orthogonal matrix.

Q. $\left|\begin{array}{cc}{x}^2+x& x\\ 3& 2\end{array}\right|+\left|\begin{array}{cc}0& -1\\ -x+1& x\end{array}\right|=\left|\begin{array}{cc}0& -2\\ 5& 1\end{array}\right|$ then x is equal to-

1. -1
2. 2
3. 1
4. No value of x