Solve Matrices Problems

Introduction to solve matrices problems:
Matrices are square and rectangular arrangements of items which are arranged in vertical and horizontal lines. Data inserted in the matrices are called entry or an element.For solve matrices problems we have to know the rows and columns of matrices.The row and column of the matrix is said to be the horizontal and vertical  lines of the matrix. Solve matrices problems are based on the four basic operations such as Addition, Subtraction, Multiplication and Transpose.

The Matrix looks like `[[3,9],[2,5]]`

Let us see how to solve the matrices problems in the following section.

Examples for Solving Matrices Problems

Let A = `[[1,2,3],[9,8,7],[2,5,3]]`         and  

      B = `[[5,3,4],[2,1,5],[9,5,1]]`

Solve: a)      A + B

            b)      A -  B

            c)      2A + 3B

            d)      A * B

            e)      AT + BT

Sol:

a) A  +  B:

For solve matrices problems, Addition can be performed only on the same order matrices. In addition, the value of the first matrix is added with the same position value of the second matrix.

            A  +  B = `[[1,2,3],[9,8,7],[2,5,3]]` + `[[5,3,4],[2,1,5],[9,5,1]]`        

                        = `[[1+5,2+3,3+4],[9+2,8+1,7+5],[2+9,5+5,3+1]]`

                       =  `[[6,5,7],[11,9,12],[11,10,4]]`

b) A  -  B:

 Sol:

Like Addition, Subtraction is done only on the same order matrix. Also the value of the first matrix is subtracted with the same position value of the second matrix

             A  -  B = `[[1,2,3],[9,8,7],[2,5,3]]`  - `[[5,3,4],[2,1,5],[9,5,1]]`

                        =  `[[1-5,2-3,3-4],[9-2,8-1,7-5],[2-9,5-5,3-1]]`   

                        =  `[[-4,-1,-1],[7,7,2],[-5,0,2]]`

c) 2A + 3B

Sol:

Step 1:

            Here first we have to calculate 2A. The number 2 multiply with all entries of the matrix A.

             2A =  `[[2*1,2*2,2*3],[2*9,2*8,2*7],[2*2,2*5,2*3]]`

                  =  `[[2,4,6],[18,16,14],[4,10,6]]` ---> 1

Step 2:

         Calculate 3B. The number 3 multiply with all entries of the matrix B.

             3B = `[[3*5,3*3,3*4],[3*2,3*1,3*5],[3*9,3*5,3*1]]`

                   = `[[15,9,12],[6,3,15],[27,15,3]]` ---> 2

Step 3:

          Now add the resultant of the step 1 and step 2.

            2A + 3B  =   `[[2,4,6],[18,16,14],[4,10,6]]` + `[[15,9,12],[6,3,15],[27,15,3]]`

                            =   `[[2+15,4+9,6+12],[18+6,16+3,14+15],[4+27,10+15,6+3]]`

                            =   `[[17,13,18],[24,19,29],[31,25,9]]`


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More Examples on Solution of Problems on Matrices

d) A * B

Sol:

         For solve matrix problems, the following condition should be satisfied in the matrix multiplication.

The column of the left matrix should be equal to the row of the second matrix.
          

            A  `xx`   B  = `[[1,2,3],[9,8,7],[2,5,3]]` * `[[5,3,4],[2,1,5],[9,5,1]]`

                         = `[[5+4+27,3+2+15,4+10+3],[45+16+63,27+8+35,36+40+7],[10+10+27,6+5+15,8+25+3]]`

                         = `[[36,20,17],[124,70,83],[47,26,36]]`

e) AT + BT

Sol:

               Transpose of the matrix is the change of the rows and columns into columns and rows respectively.

Step 1:

Find AT:

              Transpose of the matrix A is:

               AT=   `[[1,9,2],[2,8,5],[3,7,3]]`

Step 2:

Find  BT:

              Transpose of the matrix B is:  

              BT = `[[5,2,9],[3,1,5],[4,5,1]]`

Step 3:

              Now add the resultant of step 1 and step 2:

             AT  +   BT  = `[[1+5,9+2,2+9],[2+3,8+1,5+5],[3+4,7+5,3+1]]`

                            =  `[[6,11,11],[5,9,10],[7,12,4]]` .