๐Ÿ”ข Sequences, Series & Matrices
๐Ÿ”ฒ

Matrix Operations

Add, multiply, and use matrices to solve systems of equations.

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices can be added (element-wise), multiplied (row-by-column), and used to represent and solve systems of linear equations.

๐Ÿ”ข

Matrix multiplication: (AB)แตขโฑผ = sum of row i of A times column j of B. The product AB exists only when the number of columns in A equals the number of rows in B.

A=(abcd),Aโˆ’1=1adโˆ’bc(dโˆ’bโˆ’ca),detโก(A)=adโˆ’bcA = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\quad A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\quad \det(A) = ad - bc
โœ๏ธ2ร—2 Matrix Multiplication
A = [[1,2],[3,4]], B = [[5,6],[7,8]]. AB = [[(1)(5)+(2)(7), (1)(6)+(2)(8)], [(3)(5)+(4)(7), (3)(6)+(4)(8)]] = [[19, 22],[43, 50]].

Solving AX = B Using the Inverse

1Write the system as a matrix equation AX = B.
2Find det(A). If det(A) = 0, no unique solution exists.
3Compute Aโปยน.
4Multiply both sides: X = AโปยนB.
โœ๏ธ Try It!

What is the determinant of [[3, 4], [1, 2]]?

Practice with CalcVerse
Take Quiz ๐Ÿ“ โ€” 25 Questions