Algebra

Matrices & Determinants

Medium

applied

Assume $P$ , $Q$ , $R$ and $W$ are matrices of order $3 \times 3$ , $a \times 4$ , $b \times c$ and $d \times a$ respectively. If $PQ + WR$ is well defined, then the value of $ab + cd$ is:

✅ Correct Option: 3

Given matrices and their orders:

$P = 3 \times 3$

$Q = a \times 4$

$R = b \times c$

$W = d \times a$

$PQ + WR$ is well defined.

For $PQ$ to be well defined, columns of $P$ must equal rows of $Q$ :

$a = 3$

Resulting order of $PQ = 3 \times 4$

For $WR$ to be well defined, columns of $W$ must equal rows of $R$ :

$W = d \times 3$ , $R = b \times c$

$b = 3$

Resulting order of $WR = d \times c$

For $PQ + WR$ to be well defined, both matrices must have the same order:

$PQ = 3 \times 4$ , $WR = d \times c$

$d = 3$ , $c = 4$

$a = 3,\; b = 3,\; c = 4,\; d = 3$

$ab + cd = (3)(3) + (4)(3)$

$= 9 + 12$

$= 21$

2022: 7 Aug Shift 2

Let $A = (a_{ij})$ and $B = (b_{ij})$ are square matrices of same order.

(A) The number of possible matrices of order $2 \times 2$ with entries $-1, 0, 1$ is 81.

(B) $A + A'$ is skew symmetric matrix

(D) A is skew symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$

(E) $(AB)' = A'B'$

Choose the correct answer from the options given below :

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If A is a skew-symmetric matrix then, the matrix $B^TAB$ is:-

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Let $A$ be a non singular matrix of order $n \times n$ , Then $|\text{adj }(3A)|$ is equal to:

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Match List-I with List-II

List-I	List-II
Type of matrix	Conditions
(A) Square matrix A	(I) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = k & , i = j \end{cases}$ , where $k \neq 0$ is constant.
(B) Scalar Matrix A	(II) $A = [a_{ij}]_{m \times m}$
(C) Diagonal matrix A	(III) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = 1 & , i = j \end{cases}$
(D) Identity matrix A	(IV) $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0$ , $i \neq j$

Choose the correct answer from the options given below:

Assume PPP, QQQ, RRR and WWW are matrices of order 3×33 \times 33×3, a×4a \times 4a×4, b×cb \times cb×c and d×ad \times ad×a respectively. If PQ+WRPQ + WRPQ+WR is well defined, then the value of ab+cdab + cdab+cd is:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Assume $P$ , $Q$ , $R$ and $W$ are matrices of order $3 \times 3$ , $a \times 4$ , $b \times c$ and $d \times a$ respectively. If $PQ + WR$ is well defined, then the value of $ab + cd$ is: