MtxProperties

Key points

Look at the following properties that you have noticed so far while doing Addition, Subtraction and Multiplication.

1. A_2x3 + B_2x3 = C_2x3
   When we Add two matrices of the 'same order' (in this case they are 2x3) it gives another matrix of that same order (in this case a 2x3 matrix.)
2. A_2x3 - B_2x3 = C_2x3
   When we Subtract two matrices of the 'same order' (in this case they are 2x3) it gives another matrix of that same order (in this case a 2x3 matrix).
3. Rectangular Matrices;
   • A_2x3 × B_3x2 = C_2x2
     When we Multiply a matrix of order a x b by a matrix of order b x c we get a matrix of order a x c.
   • B_3x2 × A_2x4 = D_3x4
     When we Multiply a matrix of order b x a by a matrix of order a x c we get a matrix of order b x c.
     

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If you look at the above two cases, you will notice that in general
A x B is not equal to B x A.

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When do we get a square matrix as the answer?

• A_2x3 × B_3x2 = C_2x2
  When we Multiply a matrix of order a x b by a matrix of order b x a we get a square matrix of order a x a.
• A_3x3 × B_3x3 = C_3x3
  When we Multiply a matrix of order a x a by a matrix of order a x a we get a square matrix of order a x a.

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Self Assessment

Check what you have understood so far

Write the order of the matrix that you get as the result of performing the given operation with the two given matrices.(Warning! You might not be able to perform the operation with the given matrices. In that case write "No Answer".

Fill in the blank column.

First matrix	Operation	Second matrix	Resultant matrix
A3x2	+	B3x2	C?
A2x3	-	B3x2	C?
A3x2	+	B2x3	C?
A3x3	-	B3x3	C?
A2x3	x	B3x3	C?
A3x2	x	B2x4	C?
A2x3	x	B2x3	C?
A1x3	x	B3x1	C?
A2x3	x	B2x3	C?
A4x1	x	B1x2	C?
A1x2	-	B2x3	C?
A3x1	-	B1x3	C?

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Multiplication of Matrices

(1) I multiplied matrix A by matrix B. The answer was a matrix of order 2 x 3. If the order of A is 2 x 2, what could be the order of B? Think about the columns of A and the rows of B. (a) B is of order 2 x 2 Sorry! Your choice is Incorrect. Look at the key points again! (b) B is of order 2 x 3. Very good! No. of columns of A should be equal to the no. of rows in B! (c) B is of order 3 x2. Sorry! Your choice is Incorrect. Look at the key points again! (2) A is a square Matrix. What is the meaning of A2? Index rule for multiplication can be used in matrices too. (a) It is A +A. Sorry! A + A is 2A ! (b) There is no such thing like A2. Sorry! Your choice is Incorrect. Think of the multiplication ! (c) A2 = A x A. Very good! Yes. A x A can be written as A2 ! (3) P is a 2 x 3 Matrix. Can you calculate P3? Think of the necessry conditions for multiplication of matrices. (a) Yes we can. Sorry! Think about the necessary conditions for multiplication again ! (b) Yes P3 = P x P x P. Sorry! P is of order 2 x 3. Can you multiply P by a matrix of order 2 x 3? Check the Key points again. (c) We cannot calculate P3. Good! We can calculate P3 only if P is a square matrix. !

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You may try the following exercise

Additional Exercises

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