Special Types of Matrices

Definition

An identity matrix is a special type of square matrix in which all the elements of the principal diagonal (from the top left to the bottom right) are equal to 1, and all other elements are 0. It is denoted as ( I ) and is known as the multiplicative identity in matrix multiplication because any matrix multiplied by the identity matrix remains unchanged.

An identity matrix for an ( n \times n ) matrix looks like this:

[ I_n = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \ 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & 1 \end{bmatrix} ]

• The identity matrix acts like the number 1 in matrix multiplication.
• Multiplying any matrix by the identity matrix results in the original matrix.

Multiplication of an Identity Matrix by a Scalar

When an identity matrix is multiplied by a scalar, every element of the identity matrix (both the 1's on the diagonal and the 0's elsewhere) is multiplied by the scalar. This scales the matrix by the given scalar, but the overall structure of the identity matrix (with diagonal 1's) is maintained.

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Example 1:

Let ( I_2 ) be a 2x2 identity matrix, and let the scalar ( k = 4 ):

[ I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} ]

Now, multiply the identity matrix ( I_2 ) by the scalar ( k = 4 ):

[ 4 \cdot I_2 = 4 \cdot \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 4 \cdot 1 & 4 \cdot 0 \ 4 \cdot 0 & 4 \cdot 1 \end{bmatrix} ]

[ 4 \cdot I_2 = \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} ]

Thus, the result of multiplying the identity matrix by 4 is:

[ \begin{bmatrix} 4 & 0 \ 0 & 4 \end{bmatrix} ]

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Example 2:

Let ( I_3 ) be a 3x3 identity matrix, and let the scalar ( k = -2 ):

[ I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} ]

Now, multiply the identity matrix ( I_3 ) by the scalar ( k = -2 ):

[ -2 \cdot I_3 = -2 \cdot \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -2 \cdot 1 & -2 \cdot 0 & -2 \cdot 0 \ -2 \cdot 0 & -2 \cdot 1 & -2 \cdot 0 \ -2 \cdot 0 & -2 \cdot 0 & -2 \cdot 1 \end{bmatrix} ]

[ -2 \cdot I_3 = \begin{bmatrix} -2 & 0 & 0 \ 0 & -2 & 0 \ 0 & 0 & -2 \end{bmatrix} ]

Thus, the result of multiplying the identity matrix by -2 is:

[ \begin{bmatrix} -2 & 0 & 0 \ 0 & -2 & 0 \ 0 & 0 & -2 \end{bmatrix} ]

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Conclusion

• The identity matrix is a special type of matrix that has 1's along the diagonal and 0's elsewhere.
• When multiplied by a scalar, each element of the identity matrix is scaled by that scalar.
• The resulting matrix remains a diagonal matrix, but the diagonal elements are scaled by the scalar.