What is the order of $\left[ {{\rm{x\;\;y\;\;z}}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\ {\rm{h}}&{\rm{b}}&{\rm{f}}\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{x}}\ {\rm{y}}\ {\rm{z}} \end{array}} \right]?$

Question

Prepp Team · Accepted Answer

Understanding Matrix Multiplication Order

The question asks for the order (or dimension) of the resulting matrix after multiplying three given matrices: $\left[ {{\rm{x\;\;y\;\;z}}} \right]$, $\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]$, and $\left[ {\begin{array}{*{20}{c}} {\rm{x}}\\ {\rm{y}}\\ {\rm{z}} \end{array}} \right]$. To find the order of the final product, we need to perform the multiplication step-by-step and keep track of the matrix dimensions.

Determining the Dimensions of Each Matrix

First, let's identify the order of each matrix:

The first matrix is a row matrix: $\left[ {{\rm{x\;\;y\;\;z}}} \right]$. It has 1 row and 3 columns. So its order is $1 \times 3$.
The second matrix is a square matrix: $\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]$. It has 3 rows and 3 columns. So its order is $3 \times 3$.
The third matrix is a column matrix: $\left[ {\begin{array}{*{20}{c}} {\rm{x}}\\ {\rm{y}}\\ {\rm{z}} \end{array}} \right]$. It has 3 rows and 1 column. So its order is $3 \times 1$.

We can represent the multiplication as $A \times B \times C$, where:

Matrix A: $\left[ {{\rm{x\;\;y\;\;z}}} \right]$, order $1 \times 3$
Matrix B: $\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]$, order $3 \times 3$
Matrix C: $\left[ {\begin{array}{*{20}{c}} {\rm{x}}\\ {\rm{y}}\\ {\rm{z}} \end{array}} \right]$, order $3 \times 1$

Step-by-Step Matrix Multiplication

We perform matrix multiplication from left to right.

Step 1: Multiply the first two matrices (A and B)

Multiply matrix A ($1 \times 3$) by matrix B ($3 \times 3$).

For matrix multiplication $M_1 \times M_2$ to be possible, the number of columns in $M_1$ must equal the number of rows in $M_2$. If $M_1$ has order $m \times n$ and $M_2$ has order $n \times p$, the resulting matrix will have the order $m \times p$.

In this step, $M_1 = A$ (order $1 \times 3$) and $M_2 = B$ (order $3 \times 3$).

Number of columns in A is 3.
Number of rows in B is 3.

Since $3 = 3$, multiplication is possible.

The order of the resulting matrix (A $\times$ B) will be $1 \times 3$ (the number of rows from A and the number of columns from B).

Let's call the result of this multiplication matrix D. So, D = A $\times$ B, and D has order $1 \times 3$.

Step 2: Multiply the result from Step 1 (D) by the third matrix (C)

Multiply matrix D ($1 \times 3$) by matrix C ($3 \times 1$).

Now, $M_1 = D$ (order $1 \times 3$) and $M_2 = C$ (order $3 \times 1$).

Number of columns in D is 3.
Number of rows in C is 3.

Since $3 = 3$, multiplication is possible.

The order of the final resulting matrix (D $\times$ C) will be $1 \times 1$ (the number of rows from D and the number of columns from C).

Summary of Matrix Dimensions

We can summarize the process using a table showing the dimensions at each step:

Operation	Left Matrix Order	Right Matrix Order	Resulting Matrix Order
First multiplication ($A \times B$)	$1 \times 3$	$3 \times 3$	$1 \times 3$
Second multiplication ($(A \times B) \times C$ or $D \times C$)	$1 \times 3$	$3 \times 1$	$1 \times 1$

The final matrix resulting from the multiplication $\left[ {{\rm{x\;\;y\;\;z}}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{x}}\\ {\rm{y}}\\ {\rm{z}} \end{array}} \right]$ has an order of $1 \times 1$.

Conclusion on Matrix Order

The order of the final resulting matrix is $1 \times 1$.

Revision Table: Matrix Order Calculation

Step	Matrices Involved	Orders	Condition for Multiplication	Resulting Order
1	First matrix $\times$ Second matrix	$1 \times 3$ and $3 \times 3$	Inner dimensions match (3=3)	$1 \times 3$
2	Result of Step 1 $\times$ Third matrix	$1 \times 3$ and $3 \times 1$	Inner dimensions match (3=3)	$1 \times 1$

Additional Information on Matrix Dimensions and Multiplication

Understanding the order of matrices is fundamental to matrix operations. The order $m \times n$ tells us that a matrix has $m$ rows and $n$ columns.

A matrix with order $1 \times n$ is called a row matrix or row vector.
A matrix with order $m \times 1$ is called a column matrix or column vector.
A matrix with order $n \times n$ is called a square matrix.
For two matrices P (order $m \times n$) and Q (order $p \times q$) to be multiplied as PQ, the number of columns in P ($n$) must equal the number of rows in Q ($p$). The resulting matrix PQ will have the order $m \times q$.
Matrix multiplication is generally not commutative, meaning PQ is typically not equal to QP.
The multiplication $\left[ {{\rm{x\;\;y\;\;z}}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\rm{x}}\\ {\rm{y}}\\ {\rm{z}} \end{array}} \right]$ is an example of a quadratic form in matrix notation, which results in a scalar value (a $1 \times 1$ matrix).

Prepp Team · Answer

3 &times; 1

Prepp Team · Answer

1 &times; 3

Prepp Team · Answer

3 &times; 3

Operation	Left Matrix Order	Right Matrix Order	Resulting Matrix Order
First multiplication (\(A \times B\))	\(1 \times 3\)	\(3 \times 3\)	\(1 \times 3\)
Second multiplication (\((A \times B) \times C\) or \(D \times C\))	\(1 \times 3\)	\(3 \times 1\)	\(1 \times 1\)