What is a Matrix?
A matrix is a rectangular array of numbers arranged in rows and columns, used to organize data and solve systems of equations. Matrices can be added, multiplied, and inverted following specific rules.
A matrix is a grid of numbers with m rows and n columns (an m×n matrix). For a 2×2 matrix, the determinant det(A) = ad − bc tells you whether the matrix can be inverted.
- •Add corresponding entries
- •Matrices must be the same size
- •Result has the same dimensions
- •Commutative: A + B = B + A
- •Multiply rows by columns
- •Columns of A must match rows of B
- •Result size = rows(A) × cols(B)
- •Not commutative: AB ≠ BA in general
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Step-by-step worked examples
Find the determinant of [[4,3],[2,1]].
a=4, b=3, c=2, d=1 det = ad − bc = 4×1 − 3×2 det = 4 − 6 = −2
Add matrices [[1,2],[3,4]] and [[5,6],[7,8]].
Add corresponding entries [[1+5,2+6],[3+7,4+8]] = [[6,8],[10,12]]
Multiply [[1,2]] (1×2) by [[3],[4]] (2×1).
Multiply row by column: 1×3 + 2×4 = 3 + 8 = 11 (a 1×1 matrix)
Flashcards
Quick quiz
Q1.What is the determinant of [[2,5],[1,3]]?
Q2.A 2×3 matrix has how many entries?
Q3.Can a 2×3 matrix be multiplied by a 3×4 matrix?
Q4.If det(A) = 0, the matrix is...
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Common mistakes
Adding matrices of different sizes. — Correct: Matrices can only be added if they have exactly the same dimensions.
Assuming AB = BA always. — Correct: Matrix multiplication is generally not commutative — order matters.
Multiplying matrices entry-by-entry like addition. — Correct: Matrix multiplication uses row-by-column dot products, not entry-wise multiplication.
Forgetting to check dimension compatibility before multiplying. — Correct: Columns of the first matrix must equal rows of the second, or the multiplication is undefined.
FAQ
What is a matrix in math?
A rectangular grid of numbers arranged in rows and columns, used to represent linear systems and transformations.
What is the formula for a 2×2 determinant?
det(A) = ad − bc, using the entries of the matrix [[a,b],[c,d]].
What are examples of matrices?
A 2×2 identity matrix [[1,0],[0,1]] or a data table like [[85,90],[78,88]] of student scores.
How do you calculate matrix multiplication?
Multiply each row of the first matrix by each column of the second and sum the products.