Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?
Notes
Multiplying 2×2 Matrices
\(\left[{\begin{array}{cc}a & b \\c & d \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}w & x \\y & z \\\end{array} } \right]=\)\(\left[ {\begin{array}{cc}aw+by & ax+bz \\cw +dy & cx +dz \\\end{array} } \right]\)
Practice Problems
\(\textbf{1)}\) \(\left[{\begin{array}{cc}1 & 2 \\-3 & -4 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}6 & -3 \\5 & 0 \\\end{array} } \right]\)
\(\textbf{2)}\) \(\left[{\begin{array}{ccc}1 & 2 & 3\\\end{array} } \right]\)\(\left[ {\begin{array}{c}4 \\5 \\6 \\\end{array} } \right]\)
\(\textbf{3)}\) \(\left[ {\begin{array}{c}4 \\5 \\6 \\\end{array} } \right]\)\(\left[{\begin{array}{ccc}1 & 2 & 3\\\end{array} } \right]\)
\(\textbf{4)}\) \(\left[{\begin{array}{cc}1 & 2 \\3 & 4 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}5 & 6 \\7 & 8 \\\end{array} } \right]\)
\(\textbf{5)}\) \(\left[{\begin{array}{cc}1 & 1 \\1 & 0 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}2 & 2 \\3 & 3 \\\end{array} } \right]\)
\(\textbf{6)}\) \(\left[{\begin{array}{cc}0 & 1 \\1 & 0 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}4 & -5 \\1 & 0 \\\end{array} } \right]\)
\(\textbf{7)}\) \(\left[{\begin{array}{cc}0 & 1 \\1 & 0 \\\end{array} } \right]\)\(\left[ {\begin{array}{ccc}4 & -5 & 1 \\1 & 0 & 2 \\\end{array} } \right]\)
\(\textbf{8)}\) \(\left[ {\begin{array}{ccc}4 & -5 & 1 \\1 & 0 & 2 \\\end{array} } \right]\)\(\left[{\begin{array}{cc}0 & 1 \\1 & 0 \\\end{array} } \right]\)
\(\textbf{9)}\) \(\left[ {\begin{array}{ccc}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9 \\\end{array} } \right]\left[ {\begin{array}{ccc}9 & 8 & 7 \\6 & 5 & 4 \\3 & 2 & 1 \\\end{array} } \right]\)
\(\textbf{10)}\) \(\left[ {\begin{array}{ccc}1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1 \\\end{array} } \right]\left[ {\begin{array}{ccc}2 & 2 & 2 \\2 & 2 & 2 \\2 & 2 & 2 \\\end{array} } \right]\)
\(\textbf{11)}\) \(\left[ {\begin{array}{ccc}1 & 0 & 5 \\2 & 0 & 5 \\3 & 0 & 5 \\\end{array} } \right]\left[ {\begin{array}{ccc}5 & 1 & 1 \\1 & 5 & 1 \\1 & 1 & 5 \\\end{array} } \right]\)
See Related Pages\(\)
\(\bullet\text{ Intro to Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[ {\begin{array}{ccc}4 & -5 & 2 \\1 & 0 & 3 \\\end{array} } \right]\)
\(\bullet\text{ Matrix Operations}\)
\(\,\,\,\,\,\,\,\,\)\( \left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)\(+\left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)
\(\bullet\text{ Multiplying Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{cc} 1 & 2 \\ -3 & -4 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}6 & -3 \\5 & 0 \\\end{array} } \right]\)
\(\bullet\text{ Determinants}\)
\(\,\,\,\,\,\,\,\,\)\(\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=ad-bc\)
\(\bullet\text{ Cramer’s Rule}\)
\(\,\,\,\,\,\,\,\,\text{ax+by=e } \& \text{ cx+dy=f}…\)
\(\bullet\text{ Identity Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\)
\(\bullet\text{ Identity and Inverse Matrices}\)
\(\,\,\,\,\,\,\,\,A^{-1}=\displaystyle\frac{1}{ad-bc}\left[{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right]\)
\(\bullet\text{ Transpose Matrix}\)
\(\,\,\,\,\,\,\,\,\left[{\begin{array}{ccc} 1 \\ 2 \\ 5 \\ \end{array} } \right]\Rightarrow\left[{\begin{array}{c} 1 & 2 & 5 \end{array} } \right]\)
\(\bullet\text{ Rotation Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(R(\theta)=\left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right]\)
\(\bullet\text{ Eigenvectors and Eigenvalues}\)
\(\,\,\,\,\,\,\,\,(A-\lambda I)\vec{v}=\vec{0}\)
In Summary
In mathematics, a matrix is a rectangular array of numbers or expressions, arranged in rows and columns. When we multiply two matrices, we are combining them in a way that allows us to make calculations involving multiple linear transformations at once. Multiplying matrices is a fundamental operation in linear algebra.
We learn about multiplying matrices in order to understand how to apply transformations to points in space, as well as to perform other operations in
linear algebra, such as solving systems of linear equations and finding the inverse of a matrix.
Multiplying matrices may be introduced in some precalculus courses, but typically is in a linear algebra course which is usually taken by undergraduate or graduate students majoring in mathematics, engineering, or other related fields.
One common mistake when multiplying matrices is failing to check that the matrices are compatible for multiplication. In order to multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Another possible mistake is to simply multiply each entry by the corresponding entry in the other matrix.
A fun fact about multiplying matrices is that they can be used to rotate points in space around a fixed point, known as the origin. By multiplying a matrix containing the coordinates of a point by a matrix that represents a rotation, we can rotate that point around the origin.
The concept of multiplying matrices was first developed by James Joseph Sylvester, a British mathematician who lived in the 19th century.
Some related topics to multiplying matrices include matrix addition, matrix subtraction, and the determinant of a matrix. These topics are all important for understanding how to manipulate and analyze matrices in linear algebra and other related fields.
5 real world examples of Multiplying Matrices
Computer Graphics: In computer graphics, matrices are used to represent transformations such as rotations, translations, and scaling. These transformations can be combined by multiplying the corresponding matrices, allowing for the creation of complex graphics with a single matrix multiplication.
Robotics: Matrices are also used in robotics to represent the position and orientation of robots and other objects in space. By multiplying the matrices that represent the position and orientation of different parts of a robot, we can calculate the overall position and orientation of the robot as a whole.
Machine learning: In machine learning, matrices are often used to represent data sets and to perform calculations on the data.
Economics: Matrices are used in economics to represent and analyze systems of linear equations. By using matrix multiplication, we can solve systems of linear equations more efficiently than by using traditional methods.
Cryptography: In cryptography, matrices are used to encrypt and decrypt messages. One example of this is the Hill cipher.
About Andymath.com
Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.
Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.
Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!
Thank you for visiting. How exciting!