
Is matrix multiplication associative ?
Answer
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Hint: In general, the associative law of multiplication states that the order of variables will not make any difference in the result. In the given question we are going to show that matrix multiplication is associative. It means for any three matrices and , associative property states: . To prove this, let's consider three matrices in the same order.
Complete step by step answer:
We need to prove that , that is L.H.S. R.H.S.
Let , and .
L.H.S.
Here first multiply and then multiply with its result.
As per matrix multiplication, multiply each row of the first matrix with each column of the second matrix,
Multiply the values inside the bracket,
Perform the action specified in each elements,
Now multiplying with the result of ,
By multiplying each row of the first matrix with each column of the second matrix,
Perform multiplication inside the bracket,
By performing addition, we will get,
L.H.S.
R.H.S.
Whereas here, multiply first and then multiply its result with .
By performing matrix multiplication,
Performing addition,
Multiplying with ,
Again by performing matrix multiplication,
By adding the values,
R.H.S. .
The result of both and are equal.Hence we have proved that L.H.S. R.H.S., that is . Therefore matrix multiplication is associative.
Note: A matrix is a rectangular arrangement of number enclosed by a pair of brackets , in the form of horizontal and vertical lines. The numbers inside the brackets are called elements of the matrix. Matrices are usually denoted by capital letters etc. Horizontal lines are called rows and vertical lines are called columns. If a matrix has rows and columns, then the order of the matrix is said to be (read as by ) and it has elements.
The multiplication of two matrices and is only possible when the number of columns in is equal to the number of rows in . If be an order and be an order matrix, then their product has order is defined as where sum of the elements of the th row of with corresponding elements of the column of .
Associative law for addition of any three matrices: , and of the same order, say , .
The associative law for multiplication of any three matrices: and is , whenever both sides of the equality are defined.
Complete step by step answer:
We need to prove that
Let
L.H.S.
Here first multiply
As per matrix multiplication, multiply each row of the first matrix with each column of the second matrix,
Multiply the values inside the bracket,
Perform the action specified in each elements,
Now multiplying
By multiplying each row of the first matrix with each column of the second matrix,
Perform multiplication inside the bracket,
By performing addition, we will get,
L.H.S.
R.H.S.
Whereas here, multiply
By performing matrix multiplication,
Performing addition,
Multiplying
Again by performing matrix multiplication,
By adding the values,
R.H.S.
The result of both
Note: A matrix is a rectangular arrangement of number enclosed by a pair of brackets
The multiplication of two matrices
Associative law for addition of any three matrices:
The associative law for multiplication of any three matrices:
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