Question

What are the properties of a matrix?

Answer 1

Properties of a matrix:

The important properties of a matrix are:

1. Properties of matrix addition:

The matrix addition is the addition of corresponding elements of the matrices.

For the matrices $A$ and $B$,

• Commutative property: $A+B=B+A$
• Associative property: $A+\left(B+C\right)=\left(A+B\right)+C$
• Additive identity: $A+0=0+A=A$
• Additive inverse: $A+\left(-A\right)=\left(-A\right)+A=0$

2. Properties of matrix multiplication

The matrix multiplication is a product of two matrices that produce a single matrix.

For the matrices $A,B$ and $C$

• Associative property: $\left(AB\right)C=A\left(BC\right)$
• Distributive property: $A\left(B+C\right)=AB+AC$
• Multiplicative identity:$AI=IA=A$ , where $I$ is an identity matrix
• Multiplicative property of zero: $A0=0A=0$

3. Properties of scalar multiplication

If $A$ is a matrix and $k$ any constant, then the product of a constant with the matrix is equal to the product of the constant and all the elements of a matrix.

• Commutative property: $kA=Ak$
• Distributive property: $k\left(A+B\right)=kA+kB$ and $\left(k+m\right)A=kA+mA$ , where $k,m$ are scalars
• Associative property: $k\left(mA\right)=\left(km\right)A$

4. Properties of Transpose Matrix

Transpose matrix is obtained by interchanging the rows and columns. The transpose of matrix $A$ is denoted as $A^T$ or $A\text{'}$.

• ${\left(A^T\right)}^T=A$
• ${\left(A+B\right)}^T=A^T+B^T$
• ${\left(A\times B\right)}^T=B^T\times A^T$
• ${\left(kA\right)}^T=k{A}^T$

5. Properties of Inverse Matrix.

The inverse of a matrix $A$ is denoted by $A^{-1}=\frac{1}{\left|A\right|}·adj.\left(A\right)$ , where $adj.\left(A\right)$ is the adjoint matrix and $\left|A\right|$ is the determinant of the matrix $A$.

• ${\left(A^{-1}\right)}^{-1}=A$
• ${\left(A\times B\right)}^{-1}=B^{-1}\times A^{-1}$
• ${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T$