Let F(z) be the cumulative density function of the standard normal variate z, then which of the following are correct? (A) F(z) = 1 √2π ∫_-∞^z e^-…

Given:

• Matrix $A$A is a $2 \times 3$2×3 matrix (2 rows, 3 columns)
• Matrix $B$B is a $3 \times 2$3×2 matrix (3 rows, 2 columns)

When multiplying $A \times B$A×B:

Result $= (2 \times 3) \times (3 \times 2) = 2 \times 2$=(2×3)×(3×2)=2×2 matrix

So $AB$AB is a square matrix of order $2 \times 2$2×2.

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Determinants (the $| |$∣∣ symbol) only exist for square matrices.

$A$A is $2 \times 3$2×3 → not square → no determinant exists

$B$B is $3 \times 2$3×2 → not square → no determinant exists

$AB$AB is $2 \times 2$2×2 → square → determinant exists

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When a scalar $k$k multiplies an $n \times n$n×n matrix $M$M:

$$|kM| = k^n |M|$$∣kM∣=kn∣M∣

where $n$n is the order of the square matrix.

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For this problem:

$AB$AB is a $2 \times 2$2×2 matrix, so $n = 2$n=2

Scalar multiplier $k = 5$k=5

Applying the formula:

$$|5AB| = 5^2 |AB|$$∣5AB∣=52∣AB∣

$$|5AB| = 25|AB|$$∣5AB∣=25∣AB∣

Therefore, $|5AB| = 5^2|AB|$∣5AB∣=52∣AB∣.

option (A) does not satisfy the result; it is not given as correct in the NTA key, it is due to the fact that determinants $|A|$∣A∣ and $|B|$∣B∣ are not defined for non-square matrices.