# Scalar Matrix

A scalar matrix is a type of diagonal matrix. The diagonal elements of the scalar matrix are equal or same. If the elements of the scalar matrix are all equal to 1, then it becomes an identity matrix.

A square matrix A = [aij]n x n, is said to be a scalar matrix if;

• aij = 0, when i ≠ j
• aij = k, when i = j, for some constant k
Facts:

• A scalar matrix is a square matrix
• A scalar matrix is a diagonal matrix
• Every identity matrix is a scalar matrix

## Definition of Scalar Matrix

A matrix, say A = [aij]n × n is called a scalar matrix if aij = 0, when i ≠ j and aij = k, when i = j, (where k is any constant). The diagonal of the scalar matrix contains only scalar elements that are all identical.

The order of the scalar matrix is n x n. Thus, it has an equal number of rows and columns. Hence, it is also a Square matrix.

## Examples of Scalar Matrix

The examples of scalar matrix are given below:

1. Example of Scalar matrix of an order 1:
$$A=\left[\begin{array}{lllll} 1 \end{array}\right]$$
1. Example of scalar matrix of an order 2.
$$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$
1. Example of scalar matrix of an order 3.
$$\begin{bmatrix} 5 & 0 &0\\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$$
1. Example of scalar matrix of an order 4.
$$\begin{bmatrix} \sqrt{7} & 0 & 0 & 0\\ 0 & \sqrt{7} & 0 & 0 \\ 0 & 0 & \sqrt{7} & 0 \\ 0 & 0 & 0 & \sqrt{7} \end{bmatrix}$$

### Other Types of Matrices

There are majorly seven different types of matrices. They are:

1. Row matrix
2. Column matrix
3. Square matrix
4. Diagonal matrix
5. Scalar matrix
6. Identity matrix
7. Zero matrix

## Scalar Multiplication of Matrices

The scalar quantity is its original value. The properties of scalar multiplication of a matrix are defined by two matrices of the same order. Let us say, A = [aij] and B = [bij] are two matrices of the same order, say m × n. Also, the two scalars are k and l. Then the scalar multiplication are given by:

• k(A + B) = kA + kB
• (k + l)A = k A + l A

Let us find how we got the above equation.

### How to do Scalar Matrix Multiplication?

1. For k(A+) = kA + kB

LHS = k (A + B)

= k ([aij] + [bij])

= k [aij + bij]

= [k (aij + bij)]

= [(k aij) + (k bij)]

= [k aij] + [k bij]

= k [aij] + k [bij]

= kA + kB

2. For (k + l)A = k A + l A

LHS = ( k + l) A

= (k + l) [aij]

= [(k + l) aij] + [k aij] + [l aij]

= k [aij] + l [aij]

= k A + l A

Let us solve an example.

If $$A = \left(\begin{array}{cc} 2 & 3 \\ -1 & 2 \\ 1 & 0 \end{array}\right) and ~B = \left(\begin{array}{cc}3 & 1 \\ 4 & 6 \\ 8 & -2\end{array}\right)$$, then find the value of X for 2A + X = 5B.

Solution: By putting the value of matrix A and matrix B in the given equation, we get;

2A + X = 5B

$$2\left(\begin{array}{cc} 2 & 3 \\ -1 & 2 \\ 1 & 0 \end{array}\right)+X=5\left(\begin{array}{cc} 3 & 1 \\ 4 & 6 \\ 8 & -2 \end{array}\right)$$ $$2\left(\begin{array}{cc} 2 & 3 \\ -1 & 2 \\ 1 & 0 \end{array}\right)+X-2\left(\begin{array}{cc} 2 & 3 \\ -1 & 2 \\ 1 & 0 \end{array}\right)=5\left(\begin{array}{cc} 3 & 1 \\ 4 & 6 \\ 8 & -2 \end{array}\right)-2\left(\begin{array}{cc} 2 & 3 \\ -1 & 2 \\ 1 & 0 \end{array}\right)$$ $$X=5\left(\begin{array}{cc} 3 & 1 \\ 4 & 6 \\ 8 & -2 \end{array}\right)-2\left(\begin{array}{cc} 2 & 3 \\ -1 & 2 \\ 1 & 0 \end{array}\right)$$ $$=\left(\begin{array}{cc} 15 & 5 \\ 20 & 30 \\ 40 & -10 \end{array}\right)-\left(\begin{array}{cc} 4 & 6 \\ -2 & 4 \\ 2 & 0 \end{array}\right)$$ $$X=\left(\begin{array}{cc} 15-4 & 5-6 \\ 20+2 & 30-4 \\ 40-2 & -10-0 \end{array}\right)$$ $$=\left(\begin{array}{cc} 11 & -1 \\ 22 & 26 \\ 38 & -10 \end{array}\right)$$

## Practice Questions

1. Give an example of a scalar matrix of the order 3 x 3.

2. What is the order of scalar matrix $$A=\left(\begin{array}{cccc} -9 & 0 & 0 & 0 \\ 0 & -9 & 0 & 0 \\ 0 & 0 & -9 & 0 \\ 0 & 0 & 0 & -9 \end{array}\right)$$ ?

3. Multiply -5 to the matrix $$A=\left(\begin{array}{cc} 2 & 3 \\ 4 &5 \end{array}\right)$$

## Frequently Asked Questions on Scalar Matrix

### What is a scalar matrix?

A scalar matrix is a diagonal matrix that has all the elements in the diagonal equal to each other and the off-diagonal elements are zero.

### Is a diagonal matrix and scalar matrix the same?

All the scalar matrices are diagonal matrices but not all the diagonal matrices are scalar. This is because a diagonal matrix can have different elements but a scalar matrix always has the same elements.

### Is an identity matrix a scalar matrix?

An identity matrix is a scalar matrix because all its diagonal elements are equal to 1 and off-diagonal elements are 0.

### What is the order of the scalar matrix?

The order of the scalar matrix is n x n, where n is the number of rows and columns.

### Can a zero matrix be a scalar matrix?

A zero matrix is a matrix that has all its elements equal to zero. Hence, it is not a scalar matrix.