How to multiply matrices 2x2
Understand the Problem
The question is asking for the method to perform multiplication of two 2x2 matrices, which involves a specific mathematical procedure to combine them into a new matrix.
Answer
$\begin{pmatrix} a \times e + b \times g & a \times f + b \times h \\ c \times e + d \times g & c \times f + d \times h \end{pmatrix}$
Answer for screen readers
The product of the two 2x2 matrices is $\begin{pmatrix} a \times e + b \times g & a \times f + b \times h \ c \times e + d \times g & c \times f + d \times h \end{pmatrix}$
Steps to Solve
- Write down the matrices
Write the two 2x2 matrices you want to multiply. Let's denote them as Matrix $A$ and Matrix $B$.
$$ A = \begin{pmatrix} a & b \ c & d \end{pmatrix} $$ $$ B = \begin{pmatrix} e & f \ g & h \end{pmatrix} $$
- Set up the multiplication formula
To find the elements of the resulting matrix $C$, you will need to multiply rows of $A$ by columns of $B$ and sum the products.
- Calculate element $C_{11}$
The element in the first row and first column of matrix $C$ is found as follows:
$$ C_{11} = a imes e + b imes g $$
- Calculate element $C_{12}$
The element in the first row and second column of matrix $C$ is:
$$ C_{12} = a imes f + b imes h $$
- Calculate element $C_{21}$
The element in the second row and first column of matrix $C$ is:
$$ C_{21} = c imes e + d imes g $$
- Calculate element $C_{22}$
The element in the second row and second column of matrix $C$ is:
$$ C_{22} = c imes f + d imes h $$
- Combine the results into matrix $C$
Now, assemble all the calculated elements into the resulting matrix $C$:
$$ C = \begin{pmatrix} C_{11} & C_{12} \ C_{21} & C_{22} \end{pmatrix} $$
The product of the two 2x2 matrices is $\begin{pmatrix} a \times e + b \times g & a \times f + b \times h \ c \times e + d \times g & c \times f + d \times h \end{pmatrix}$
More Information
Multiplying matrices involves combining rows of the first matrix with columns of the second matrix, producing a new matrix.
Tips
Common mistakes include forgetting to sum the products of the corresponding elements and multiplying rows by rows or columns by columns instead of rows by columns.
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