Let A, B, C, D and E be matrices of order $2 \times n, 3 \times k, 2\times p, n \times 3$ and $p \times k$ respectively. Choose the correct statement(s) from the following?

(A) EB + DB will be defined if $k = 3, p = n$.

(B) EB + DB will be defined if $k = 2, p = 3$.

(C) If n = p = 2, then the order of the matrix $5A^2 - 3C$ is $2 \times 2$.

(D) If n = p, then the order of the matrix $5A^2 - 3C$ is $p \times k$.

Choose the correct answer from the options given below:

Let me break down each statement so it's crystal clear!

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Understanding the given matrices:

$A: 2 \times n$

$B: 3 \times k$

$C: 2 \times p$

$D: n \times 3$

$E: p \times k$

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Key Rules to Remember:

For matrix multiplication $XY$ to be defined: Number of columns in $X =$ Number of rows in $Y$

If $X$ is $m \times r$ and $Y$ is $r \times n$, then $XY$ is $m \times n$

For matrix addition $X + Y$ to be defined: Both matrices must have exactly the same dimensions

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Checking Statement (A): $EB + DB$ will be defined if $k = 3, p = n$

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Finding $EB$:

$E$ is $p \times k$ and $B$ is $3 \times k$

For $EB$ to work: columns of $E =$ rows of $B$

Columns of $E = k$, Rows of $B = 3$

So we need $k = 3$

Result: $EB$ is $p \times 3$

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Finding $DB$:

$D$ is $n \times 3$ and $B$ is $3 \times k$

Columns of $D = 3$, Rows of $B = 3$ (matches!)

If $k = 3$, then $DB$ is $n \times 3$

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Can we add $EB + DB$?

$EB$ is $p \times 3$

$DB$ is $n \times 3$

For addition: we need $p = n$ (given in statement!)

Both become $n \times 3$

Statement (A) is CORRECT

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Checking Statement (B): $EB + DB$ will be defined if $k = 2, p = 3$

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Finding $EB$:

$E$ is $p \times k = 3 \times 2$ and $B$ is $3 \times k = 3 \times 2$

Columns of $E = 2$, Rows of $B = 3$

$2 \neq 3$

$EB$ is NOT defined!

Since $EB$ itself can't be calculated, $EB + DB$ is impossible.

Statement (B) is INCORRECT

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Checking Statement (C): If $n = p = 2$, then the order of $5A^2 - 3C$ is $2 \times 2$

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Finding $A^2$:

$A$ is $2 \times n = 2 \times 2$

For $A^2 = A \times A$: columns of $A =$ rows of $A$

Both are $2$

$A^2$ is $2 \times 2$

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$5A^2$ is $2 \times 2$ (scalar multiplication doesn't change dimensions)

$C$ is $2 \times p = 2 \times 2$

$3C$ is $2 \times 2$

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Calculating $5A^2 - 3C$:

Both are $2 \times 2$

Result is $2 \times 2$

Statement (C) is CORRECT

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Checking Statement (D): If $n = p$, then the order of $5A^2 - 3C$ is $p \times k$

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Finding $A^2$:

$A$ is $2 \times n$

For $A^2$: columns of $A =$ rows of $A$

Columns $= n$, Rows $= 2$

This only works if $n = 2$

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What if $n \neq 2$?

If $n = p = 5$ (for example), $A$ is $2 \times 5$

$A^2$ is undefined!

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Even if it works (when $n = 2$):

From Statement (C), the result is $2 \times 2$

The statement claims it's $p \times k$

That would mean $p = 2$ and $k = 2$, but $k$ is independent!

The dimension doesn't depend on $k$ at all

Statement (D) is INCORRECT

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Statements (A) and (C) are correct.