Adding Identity Matrix to Matrix: Is 5 a Scalar?

In summary, adding a scalar to a matrix is not a defined operation and is meaningless. The identity matrix is a square matrix with ones along the main diagonal and zeros everywhere else. It is used to represent scalar multiplication in linear algebra. However, it is not possible to add a scalar and a matrix. Similarly, the concept of "zero" is different for a scalar and a matrix. While a scalar zero is simply the number zero, a matrix zero (also known as the zero matrix) is a matrix filled with zeros. These distinctions are important in matrix operations and cannot be interchanged.
  • #1
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Say i have a matrix ,

[tex]\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5[/tex]

is it correct if i do it this way ,

[tex]\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}[/tex]

[tex]=\begin{bmatrix}{9}&{3}\\{-1}&{12}\end{bmatrix}[/tex]

is 5 a scalar = 5I where I is an identity matrix ?
 
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  • #2


yes, I think you are correct
 
  • #3


but one problem, how can you make a prediction about the size of that matrix as 2 x 2. I can say that the unit matrix is of 3 x 3 or 4 x 4 and then can you add?
 
  • #4


snshusat161 said:
but one problem, how can you make a prediction about the size of that matrix as 2 x 2. I can say that the unit matrix is of 3 x 3 or 4 x 4 and then can you add?

thanks , how if we assume that the identity matrix is a 2x2 matrix ? since the given matrix is also a 2x2

btw , i have this question ,

is 0 = [tex]\begin{bmatrix}{0}&{0}\\{0}&{0}\end{bmatrix}[/tex] (zero matrix)


or 2 = [tex]\begin{bmatrix}{2}&{2}\\{2}&{2}\end{bmatrix}[/tex]
 
  • #5


answer is yes for first, in the case of multiplication you can consider but when have to add you can't do it because you can only add it when both the matrix is of same order. To multiply any matrix with any scalar you have to multiply every term inside the matrix by that scalar.
 
  • #6


Also I'm not sure whether 2 = [tex]
\begin{bmatrix}{2}&{2}\\{2}&{2}\end{bmatrix}
[/tex]

In my view it is wrong to write.
 
  • #7


thereddevils said:
Say i have a matrix ,

[tex]\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5[/tex]

is it correct if i do it this way ,

[tex]\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}[/tex]

[tex]=\begin{bmatrix}{9}&{3}\\{-1}&{12}\end{bmatrix}[/tex]

is 5 a scalar = 5I where I is an identity matrix ?
No, in general the sum of a vector or matrix and a scalar is simply not defined. "A+ 5" where A is a matrix makes no sense. Writing "5" or any other scalar to indicate a matrix is very bad notation. If it is intended to be interpreted as "A+ 5I" then it should be written that way.
 
  • #8


It would be better to write 0 = [tex]\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}[/tex] where I have specifically used bold face for the "0" to indicate it is NOT a scalar but a matrix.

In general, the 0 matrix is NOT equal to the 0 scalar. A scalar is not, and cannot be equal to, a matrix.
 
  • #9


I need verification for my view from another members here. I think matrix is not any number like determinant. It is only a set of data so it is meaningless to add or subtract it by any scalar cause addition and subtraction is only done between two quantities having same dimension. Like you can't add velocity and displacement similarly you can't add matrix and scalar. One can add determinant with scalar cause it is also a number which can be found upon its solution.
 
  • #10


thanks all for helping , i got it !
 
  • #11


snshusat161 said:
I need verification for my view from another members here. I think matrix is not any number like determinant. It is only a set of data so it is meaningless to add or subtract it by any scalar cause addition and subtraction is only done between two quantities having same dimension. Like you can't add velocity and displacement similarly you can't add matrix and scalar. One can add determinant with scalar cause it is also a number which can be found upon its solution.
Right -- a matrix and a number are different. As HallsOfIvy said in his post, it is meaningless to add a scalar (number) and a matrix.

The addition problem posed in the OP is meaningless.
 

1. What is the identity matrix?

The identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. It is denoted by the symbol "I" or "In" where "n" represents the number of rows and columns.

2. What is the purpose of adding the identity matrix to a matrix?

The purpose of adding the identity matrix to a matrix is to preserve the original matrix. This means that the resulting matrix will have the same values as the original matrix, but with the identity matrix added to it.

3. How do you add the identity matrix to a matrix?

To add the identity matrix to a matrix, you simply add the corresponding elements of the two matrices. For example, if the matrices have the same dimensions, you would add the elements in the first row and first column, then the second row and second column, and so on.

4. Is 5 a scalar when adding the identity matrix to a matrix?

No, 5 is not a scalar when adding the identity matrix to a matrix. A scalar is a single number, while 5 is a matrix with only one element. The identity matrix is a matrix of 1s and 0s, not a number.

5. What is the result of adding the identity matrix to a matrix?

The result of adding the identity matrix to a matrix is a new matrix with the same dimensions as the original matrix. However, the values in the resulting matrix will be the sum of the corresponding elements in the two matrices. The resulting matrix will have the same values as the original matrix, but with 1s along the main diagonal and 0s everywhere else.

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