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Matrix Multiplication

  • Matrix Multiplication

  • Scalar Multiplication

  • GATE Previous Year Questions

Matrix Multiplication

Suppose a vector V is transformed by matrix M1  to generate vector VT.

GATE: Matrix Algebra

​Further VT is transformed by M2  to generate vector VTT.

GATE: Matrix Algebra

​If V is transformed by M  to generate vector VTT directly.

GATE: Matrix Algebra

Here transformation M is same as combined effect of transformation Mperformed after transformation M1. M is referred as multiplication of matrices Mand M.

GATE: Matrix Algebra

"Multiplication of matrices is nothing but one transformation after another from right to left."

Learn Basics: Matrix as linear transformation of Vectors

Suppose matrix Mand M are:

GATE: Matrix Algebra
GATE: Matrix Algebra

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

GATE: Matrix Algebra
GATE: Matrix Algebra

iT and jT represent first and second column respectively of overall transformation matrix i.e. product matrix of M1 and M2.

GATE: Matrix Algebra

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.

GATE: Matrix Algebra

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.

GATE: Matrix Algebra
  • Matrix multiplication is associative.

GATE: Matrix Algebra
  • Transpose of product shows following relation:

GATE: Matrix Algebra

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

Solution: Option A is the correct answer.

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