Multiplication of Matrix with a Vector can be done in two ways i.e. Inner Product and Outer Product.

**Matrix-Vector Inner Product**: In algebra, a vector is represented by single column matrix e.g. below given single column matrix is a vector.

So, we can say any matrix A with dimensions of m*n is a bunch of m*1 column vectors lined up next to each other.

**Matrix Vector Inner Product**: Assume that **a** and **b** are vectors, each with the same number of elements. Then, the **inner product** of **a** and **b** is s.

**a’b** = **b’a** = s

where

**a** and **b** are column vectors, each having n elements,

**a’** is the transpose of **a**, which makes **a’** a row vector,

**b’** is the transpose of **b**, which makes **b’** a row vector, and

s is a scalar; that is, s is a real number – not a matrix.

Note this interesting result. The product of two matrices is usually another matrix. However, the inner product of two vectors is different. It results in a real number – not a matrix. This is illustrated below.

Then, **a*b = **21 * 42 + 24 *62 + 33 * 27 = 3261**.**

Note: The inner product is also known as the **dot product** or as the **scalar product.**

**Matrix- Vector Outer Product: **Assume that **a** and **b** are vectors. Then, the **outer product** of **a** and **b** is **C**.

**ab’**= **C**

where

**a** is a column vector, having *m* elements,

**b** is a column vector, having *n* elements,

**b’** is the transpose of **b**, which makes **b’** a row vector, and

**C** is a rectangular *m* x *n* matrix

Unlike the inner product, the outer product of two vectors produces a rectangular matrix, not a scalar. This is illustrated below.

Notice that the elements of Matrix **C** consist of the product of elements from Vector **A** crossed with elements from Vector **B**. Thus, Matrix **C** winds up being a matrix of cross products from the two vectors.