Some important property of Matrix Multiplication

 Multiplicative identity property

The $n\times n$n, times, n identity matrix, denoted $I_n$I, start subscript, n, end subscript, is a matrix with $n$n rows and $n$n columns. The entries on the diagonal from the upper left to the bottom right are all $1$1's, and all other entries are $0$0.

For example:

$I_2=\left[\begin{array}{rr}{1} &0 \\ 0& 1 \end{array}\right]\quad I_3=\left[\begin{array}{rr}{1} &0 &0 \\ 0& 1&0\\0&0&1 \end{array}\right]\quad I_4=\left[\begin{array}{rr}{1} &0 &0&0 \\ 0& 1&0&0\\0&0&1&0\\0&0&0&1 \end{array}\right]$

The multiplicative identity property states that the product of any $n\times n$n, times, n matrix $A$A and $I_n$I, start subscript, n, end subscript is always $A$A, regardless of the order in which the multiplication was performed. In other words, $A\cdot I=I\cdot A=A$

Multiplicative property of zero

A zero matrix is a matrix in which all of the entries are $0$0. For example, the $3\times 3$3, times, 3 zero matrix is $O_{3\times 3}=\left[\begin{array}{rrr}0 & 0&0 \\ 0 & 0&0 \\ 0 & 0&0 \end{array}\right]$.

A zero matrix is indicated by $O$O, and a subscript can be added to indicate the dimensions of the matrix if necessary.

The multiplicative property of zero states that the product of any $n\times n$n, times, n matrix and the $n\times n$n, times, n zero matrix is the $n\times n$n, times, n zero matrix. In other words, $A\cdot O=O\cdot A=O$A, dot, O, equals, O, dot, A, equals, O.

The dimension property

One property that is unique to matrices is the dimension property. This property has two parts:

(m×n).(n×k)=(m×k)

1.The product of two matrices will be defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.

2. If the product is defined, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

For example, if $A$A is a $\blueD3\times \maroonC2$start color #11accd, 3, end color #11accd, times, start color #ed5fa6, 2, end color #ed5fa6 matrix and if $B$B is a $\maroonC2\times \goldD4$start color #ed5fa6, 2, end color #ed5fa6, times, start color #e07d10, 4, end color #e07d10 matrix, the dimension property tells us:

• The product $AB$A, B is defined.
• $AB$A, B will be a $\blueD3\times \goldD4$start color #11accd, 3, end color #11accd, times, start color #e07d10, 4, end color #e07d10 matrix.

A, dot, I, equals, I, dot, A, equals, A.