Determinant of Sum of Matrices

Introduction

The determinant of sum of matrices represents that the additon process on the matrices of determinant or adding the two matrices first and then we calculate the determinant. The matrix determinant is a special number associated with any square matrix. The meaning of the determinant is the scale factor for the measure when the matrix is regarded as a linear transformation.

(Source from Wikipedia)

Examples to Explain “determinant of Sum of Matrices”

  • Let us consider two matrices   `|[3,1,-1],[2,-2,0],[1,2,-1]|`   and  `|[12,13,1],[14,15,1],[1,1,1]|`   in the determinant form and perform the operation of addition.

Solution:

Let us consider the determinant  Q = `|[3,1,-1],[2,-2,0],[1,2,-1]|`    W = `|[12,13,1],[14,15,1],[1,1,1]|`

Perform the addition operation

Q + W =  `|[3,1,-1],[2,-2,0],[1,2,-1]|`    +   `|[12,13,1],[14,15,1],[1,1,1]|`

Q + W =  `|[3+12,1+13,-1+1],[2+14,-2+15,0+1],[1+1,2+1,-1+1]|`

Q + W = `|[15,14,0],[16,13,1],[2,3,0]|`

  • Let us consider two matrices   `[[3,1,-1],[2,-2,0],[1,2,-1]]`   and  `[[12,13,1],[14,15,1],[1,1,1]]`   and perform the operation of addition after that calculate the determinant value.

Solution:

Let us consider the matrices  Q = `[[3,1,-1],[2,-2,0],[1,2,-1]]`  W = `[[12,13,1],[14,15,1],[1,1,1]]`

We have to calculate | Q +  W |

Perform the addition operation

Q + W =  `[[3,1,-1],[2,-2,0],[1,2,-1]]`     +   `[[12,13,1],[14,15,1],[1,1,1]]`

Q + W =  `[[3+12,1+13,-1+1],[2+14,-2+15,0+1],[1+1,2+1,-1+1]]`

Q + W = `[[15,14,0],[16,13,1],[2,3,0]]`

Let us calculate the determinant for the resulted matrix after sum of two given matrices

| Q + W | = 15(0-3) – 14(0-26) +0(48-26)

| Q + W | = -45 + 364

| Q + W | = 319

Problems to Explain “determinant of Sum of Matrices”

  • Let us consider two matrices   `|[3,1,-1],[2,-2,0],[1,2,-1]|`   and  `|[12,13,1],[14,15,1],[1,1,1]|`   in the determinant form and perform the operation of addition after calculating its determinant values.

Solution:

We have to calculate | Q |  + | W |

Let us consider the determinant  Q = `|[3,1,-1],[2,-2,0],[1,2,-1]|`    W = `|[12,13,1],[14,15,1],[1,1,1]|`

| Q | = 3(2-0) – 1(-2+0) – 1(4+2)

| Q | =6 +2 – 6

| Q | = 2

| W | =12(15-1) – 13(14-1) +1(14-15)

| W | =168 – 169 -1

| W | = -2

| Q |  + | W | = 2 – 2 = 0

On the above problems we have different answers on different operations.

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