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Foundations of Matrices
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Introduction to the concept of matrices, their properties, and basic matrix operations.
Want a deeper conceptual understanding? Try our interactive lesson!
Matrix notation
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A matrix is simply a rectangular array of numbers written in rows and columns. For example, the matrix
has 2 rows and 3 columns. We can reference a specific entry using the notation Mi,j where i is the row and j the column. For example, M1,3 is the first row and thirs column, which is π.
Order of a matrix
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The order of a matrix denotes its dimensions. We say a matrix has order n×m (read "n by m") to denote that it has n rows and m columns. For example, the matrix
has order 2×3.
Adding & subtracting matrices
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When matrices have the same order m×n, we can add or subtract them by simply adding or subtracting the components:
(1324)+(5768)=(1+53+72+64+8)=(610812)
Scalar multiplication of a matrix
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We can multiply a matrix by a scalar by simply multiplying each entry by a scalar:
2(1324)=(2×12×32×22×4)=(2648)
Matrix multiplication
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If A has order m×n and B has order n×p, then the product AB is defined and will have order m×p.
For matrix multiplication in general, the number of columns in the first matrix must equal the number of rows in the second matrix.
Each entry of AB is the dot product of a row of A with a column of B.
For example, if we multiply a 2×2 matrix A by a 2×3 matrix B, the product will be 2×3:
(1324)(5869710) =(1⋅5+2⋅83⋅5+4⋅81⋅6+2⋅93⋅6+4⋅91⋅7+2⋅103⋅7+4⋅10) =(214724542761)
Matrix multiplication is not commutative
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In general, for matrices A and B,
AB=BA
Note: this does not mean AB is never equal to BA, just that it is not in general.
Matrix multiplication is associative
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For all matrices A,B and C, it is true that
A(BC)=(AB)C
Matrix multiplication is distributive
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For all matrices A,B and C, it is true that
A(B+C)=AB+AC
Zero Matrix
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A matrix with all 0 entries is called the "zero" matrix, regardless of order. We denote it 0.
0=(000000)
Matrix math with technology
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When you have matrices with only numbers and no parameters (eg k), you can add, multiply, and find powers of matrices using your calculator.
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