Adding Identity Matrix to Matrix: Is 5 a Scalar?

[thereddevils]

Say i have a matrix ,

$$\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5$$

is it correct if i do it this way ,

$$\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}$$

$$=\begin{bmatrix}{9}&{3}\\{-1}&{12}\end{bmatrix}$$

is 5 a scalar = 5I where I is an identity matrix ?

 

[snshusat161]

yes, I think you are correct

 

[snshusat161]

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?

 

[thereddevils]

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 = $$\begin{bmatrix}{0}&{0}\\{0}&{0}\end{bmatrix}$$ (zero matrix)


or 2 = $$\begin{bmatrix}{2}&{2}\\{2}&{2}\end{bmatrix}$$

 

[snshusat161]

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.

 

[snshusat161]

Also I'm not sure whether 2 = $$\begin{bmatrix}{2}&{2}\\{2}&{2}\end{bmatrix}$$

In my view it is wrong to write.

 

[HallsofIvy]

Say i have a matrix ,

$$\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5$$

is it correct if i do it this way ,

$$\begin{bmatrix}{4}&{3}\\{-1}&{7}\end{bmatrix}+5\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}$$

$$=\begin{bmatrix}{9}&{3}\\{-1}&{12}\end{bmatrix}$$

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.

 

[HallsofIvy]

It would be better to write 0 = $$\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$$ 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.

 

[snshusat161]

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.

 

[thereddevils]

thanks all for helping , i got it !

 

[Mark44]

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.