If P, Q and R are matrices of order 2x3, 3x5 and 5x3 respectively. Then which of the following are valid?

(A) P Q R

(B) P R Q

(C) Q R

(D) R Q

(E) P R

Choose the correct answer from the options given below:

Matrix P: 2×3 (2 rows, 3 columns)

Matrix Q: 3×5 (3 rows, 5 columns)

Matrix R: 5×3 (5 rows, 3 columns)

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

If Matrix A is $m \times n$ and Matrix B is $p \times q$, then $A \times B$ is valid when $n = p$, with result being $m \times q$.

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(A) P Q R

Can P×Q be computed?

P is $2 \times 3$, Q is $3 \times 5$

Columns of P (3) = Rows of Q (3)

$P \times Q$ gives a $2 \times 5$ matrix

Can (PQ)×R be computed?

$PQ$ is $2 \times 5$, R is $5 \times 3$

Columns of PQ (5) = Rows of R (5)

Final result: $2 \times 3$ matrix

Valid

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(B) P R Q

Can P×R be computed?

P is $2 \times 3$, R is $5 \times 3$

Columns of P (3) ≠ Rows of R (5)

Invalid

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(C) Q R

Q is $3 \times 5$, R is $5 \times 3$

Columns of Q (5) = Rows of R (5)

Result: $3 \times 3$ matrix

Valid

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(D) R Q

R is $5 \times 3$, Q is $3 \times 5$

Columns of R (3) = Rows of Q (3)

Result: $5 \times 5$ matrix

Valid

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(E) P R

P is $2 \times 3$, R is $5 \times 3$

Columns of P (3) ≠ Rows of R (5)

Invalid

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Valid expressions: (A), (C), and (D)