Triangular Matrix

Q.

What is a$2\times 3$ matrix?

Q. Find the missing term that should be placed in place of ? for a logical pattern.

1. $70$
2. $80$
3. $90$
4. $100$

Q. Let S be the set containing all $3\times 3$ matrices with entries from {$–1,0,1$}. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is $6$ is ​​​​​​​

Q. What is the number of all possible matrices of order $2\times 3$ with each entry $0,1$?

Q. For the matrices $A$ and $B$, verify that $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$ where:
$\left(i\right)$ $A=\left[\begin{array}{c}1\\ -4\\ 3\end{array}\right],B=\left[\begin{array}{ccc}-1& 2& 1\end{array}\right]$
$\left(ii\right)$ $A=\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right],B=\left[\begin{array}{ccc}1& 5& 7\end{array}\right]$

Q. Diagonal matrix is also a triangular matrix how ? And what is triangular matrix...?

Q. What is the trace of matrix A=$\left(\begin{array}{ccc}5& 5& 8\\ 9& 8& 4\\ 7& 2& 3\end{array}\right]$?

1. 15
2. 20
3. 14
4. 16

Q. The value of k for which the Planes x-y+z+1=0, Kx+3y+2z-3=0 and 3x+ky+z-2=0 form a triangular prism must be equal to

Q. Square matrix $[a_{ij}{]}_{n\times n}$ will be an upper triangular matrix, if

1. $a_{ij}=0$, for i<j
2. $a_{ij}\ne 0$, for i>j
3. $a_{ij}=0$, for i>j
4. $Noneofthese$

Q. Addition of two matrices of order $3\times 3$ results in a matrix of order

1. $2\times 2$
2. $2\times 3$
3. $3\times 2$
4. $3\times 3$

Q.

Assume X, Y, Z, W and P are matrices of order, and respectively. The restriction on n, k and p so that will be defined are:

A. k = 3, p = n

B. k is arbitrary, p = 2

C. p is arbitrary, k = 3

D. k = 2, p = 3

Q.

Construct a $3\times 4$ matrix whose elements are given by
$\left(i\right)a_{ij}=\frac{1}{2}|-3i+j|$

$\left(ii\right)a_{ij}=2i-j$.

Q. If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.

Q.

Construct a 2 × 2 matrix, , whose elements are given by:

(i)

(ii)

(iii)

Q. A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?

Q.

If possible, find the sum of the matrices A and B, where:

A = $\left[\begin{array}{c}\sqrt{3}1\\ 23\end{array}\right]$ and B = $\left[\begin{array}{c}xyz\\ abc\end{array}\right]$

Q. If A is a square matrix of order 3 with determinant 4, then write the value of |−A|.

Q. If A and B are square matrices of the same order such that |A| = 3 and AB = I, then write the value of |B|.

Q. If A is a matrix of order 3 and |A| = 8, then |adj A| =
(a) 1
(b) 2
(c) 2³
(d) 2⁶

Q. Let A = [a_ij] be a square matrix of order 3 × 3 and C_ij denote cofactor of a_ij in A. If |A| = 5, write the value of a₃₁ C₃₁ + a₃₂ C₃₂ a₃₃ C₃₃.

Q. If $A$ is a $3$rd order square diagonal matrix of non-positive entries such that $A^2=I$, then

1. There may exist some diagonal elements such that Tr$\left(A\right)$ is zero
2. Value of $|A|$ is $-1$
3. $A^{-1}$ does not exist
4. $|3\text{Adj}\left(2A\right)|=1728$

Q. What is the trace of matrix A=$\left(\begin{array}{ccc}5& 5& 8\\ 9& 8& 4\\ 7& 2& 3\end{array}\right]$?

1. 15
2. 20
3. 14
4. 16

Q. Let k be a positive real number and let $A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$ .

1. $4$
2. $5$
3. $6$
4. $3$

Q. If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
(a) dⁿ
(b) dⁿ⁻¹
(c) dⁿ⁺¹
(d) d

Q. The value of $\text{Adj}\left(\text{Adj}A\right)$ is equal to

1. $A$
2. $8A$
3. $-A$
4. $16A$

Q. Construct a $3\times 4$ matrix $A=\left[a_{ij}\right]$, whose element $a_{ij}$ is $a_{ij}=i+j$.

Q. Determine whether given matrices product is defined or not. If the product is defined, state the dimension of the product matrix.

Q. Assertion :$\left[\begin{array}{ccc}3& 0& 0\\ 0& 4& 0\\ 0& 0& 7\end{array}\right]$ is a diagonal matrix. Reason: If $A=\left[a_{ij}\right]$ is a square matrix such that $a_{ij}=0,\forall i\ne j,$ then $A$ is called a diagonal matrix.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. If A is a matrix of order m × n and B is a matrix such that AB^T and B^TA are both defined, then the order of matrix B is

(a) m × n
(b) n × n
(c) n × m
(d) m × n

Disclaimer: option (a) and (d) both are the same.

Q. The restriction on $n,k$ and $p$ so that $PY+WY$ will be defined are:

1. $k=3,p=n$
2. $k=2,p=3$
3. $k$ is arbitary, $p=2$
4. $p$ is arbitary, $k=3$