**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.