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Matrix Multiplication

Matrix Multiplication

Scalar Multiplication

GATE Previous Year Questions
Matrix Multiplication
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Suppose a vector V is transformed by matrix M1 to generate vector VT.
â€‹Further VT is transformed by M2 to generate vector VTT.
â€‹If V is transformed by M to generate vector VTT directly.
Here transformation M is same as combined effect of transformation M2 performed after transformation M1. M is referred as multiplication of matrices M1 and M2 .
â€‹"Multiplication of matrices is nothing but one transformation after another from right to left."
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Learn Basics: Matrix as linear transformation of Vectors
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Suppose matrix M1 and M2 are:
First and second column of matrix M1 represents transformation of unit vector i and j respectively. Further when transformation M2 is performed, overall effect of the two will change the unit vectors to
iT and jT represent first and second column respectively of overall transformation matrix i.e. product matrix of M1 and M2.
Scalar Multiplication of a Matrix
If a matrix [M] is multiplied by a scalar k, all the elements of the matrix get multiplied.
This way, direction of transformation remain same but magnitude gets multiplied by k times.
Note:

Matrix multiplication M1M2 is possible only if number of column in matrix M1 is equal to number of rows in matrix M2.

Multiplication of matrix is not commutative, since applying transformation M1 after M2 is not same as applying transformation M2 after M1.

Matrix multiplication is associative.

Transpose of product shows following relation:
Example:
GATE 2020: Multiplication of real valued square matrices of same dimensions is
a) Associative
b) Commutative
c) always positive definite
d) not always possible to compute
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Solution: Option A is the correct answer.
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