Matrix Definitions

The matrix A has m rows and n columns.

The order of A is m × n.

Addition and Subtraction of Matrices

For any m × n matrices A and B,

A + B = [a_ij + b_ij]
A - B = [a_ij - b_ij]

Scalar Multiplication of Matrices

For any m × n matrix A and scalar k,

kA = [ka_ij]

Associative and Commutative Properties of Addition and Scalar Multiplication

For any m × n matrices A, B and C and scalars k and l,

A + B = B + A
A + (B + C) = (A + B) + C
(kl)A = k(lA)

Additive Inverse

For any m × n matrix A, there exists a unique matrix -A such that

A + (-A) = 0,

where 0 is the  m × n matrix [a_ij], where a_ij=0 for every i and j.

Distributive Property

For any m × n matrices A and B and scalars k and l,

k(A + B) = kA + kB
(k + l)A = kA + lA

Matrix Multiplication

For any m × n matrix A, and n × p matrix B,

C = AB is the m × p matrix [c_ik], where c_ik = a_i1b_1k + a_i2b_2k + ... + a_inb_nk

In other words, to find the value of cell c_ik of the product matrix, C, you sum the n individual products of cells in the row i of matrix A and column k of matrix B.

Note: Matrix multiplication is not commutative.  In other words, in general, AB ¹ BA

Associative and Distributive Properties of Matrix Multiplication

For matrices A, B, and C, assuming the indicated operations are possible:

A(BC) = (AB)C
A(B + C) = AB + AC
(B + C)A = BA + CA

Reminder: The commutative property does not hold for matrix multiplication.

Augmented Matrix

Consider the set of m simultaneous linear equations in m variables v₁, v₂, ... v_m,

a₁₁v₁ + a₁₂v₂ + ... + a_1mv_m = c₁
a₂₁₁ + a₂₂₂ + ... + a_2mm = c_2
...
a_m11 + a_m22 + ... + a_mmm = c_m

The coefficients and constants can be put into an m × n augmented matrix A by setting n = m+1, and letting a_in=c_i.

Matrix Rank

The matrix rank is the maximum number of linearly independent rows or columns.

Matrix rank is invariant under row or column interchange.

For an m × n matrix A, rank [A] £ min(m,n).

For an n × n matrix A, rank [A] = n iff |A| ¹ 0.

To calculate matrix rank by hand, convert a matrix to its RREF -- Reduced Row Echelon Form -- and count the number of non-zero rows.

Determinant

The determinant of a square matrix (usually denoted by writing the matrix between vertical bars) is the algebraic sum of all possible products where the number of factors in each product is the same as the number of rows or columns, each factor in a given product is taken from a different row and column, and the sign of a product is positive or negative depending upon whether the number of permutations necessary to place the indices representing each factor's position in its row or column in the order of the natural numbers is odd or even.

Example: consider this 4x4 matrix:

₄P₄=24 (there are 24 permutations of four things taken four at a time) so there are 24 products where each factor in a given product is taken from a different row and column.  If the four products are taken from successive columns and rows 1, 2, 3, 4, the product is added to find the determinant.  If two rows are interchanged (e.g. rows 1, 2, 4, 3) then the product is subtracted to find the determinant.  Another way to look at it is this: If an even number of operations is needed to restore the rows to 1, 2, 3, 4 order, then the product is added to find the determinant.  If an odd number of operations is needed to restore the rows to 1, 2, 3, 4 order, then the product is subtracted to find the determinant.

In each product, below, the four factors are taken from successive columns.  If the order of the rows was achieved by an even number of pairwise interchanges, then a "plus" sign is attached to it.  If the order of the rows was achieved by an odd number of pairwise interchanges, then a "minus" sign is attached to it:

|A| = 

a₁₁a₂₂a₃₃a₄₄
-a₁₁a₂₂a₃₄a₄₃
+a₁₁a₂₃a₃₄a₄₂
-a₁₁a₂₃a₃₂a₄₄
+a₁₁a₂₄a₃₂a₄₃
-a₁₁a₂₄a₃₃a₄₂
-a₁₂a₂₃a₃₄a₄₁
+a₁₂a₂₃a₃₁a₄₄
-a₁₂a₂₄a₃₁a₄₃
+a₁₂a₂₄a₃₃a₄₁
-a₁₂a₂₁a₃₃a₄₄
+a₁₂a₂₁a₃₄a₄₃
+a₁₃a₂₄a₃₁a₄₂
-a₁₃a₂₄a₃₂a₄₁
+a₁₃a₂₁a₃₂a₄₄
-a₁₃a₂₁a₃₄a₄₂
+a₁₃a₂₂a₃₄a₄₁
-a₁₃a₂₂a₃₁a₄₄
-a₁₄a₂₁a₃₂a₄₃
+a₁₄a₂₁a₃₃a₄₂
-a₁₄a₂₂a₃₃a₄₁
+a₁₄a₂₂a₃₁a₄₃
-a₁₄a₂₃a₃₁a₄₂
+a₁₄a₂₃a₃₂a₄₁

To find the determinant of larger and larger matrices, you would need to contend with faster-than-exponential growth of the number of permutations, if you want to carry out the calculations as I have described thus far.  So it is important to realize that the determinant of a matrix [a_ij] is equal to

a₁₁ multiplied by the determinant of the matrix containing all but the row and column containing a₁₁,

minus a₁₂ multiplied by the determinant of the matrix containing all but the row and column containing a₁₂,

plus a₁₃ multiplied by the determinant of the matrix containing all but the row and column containing a₁₃,

etc.

This suggests a recursive algorithm for finding the determinant of matrix A of order n:

If n==1 then return a₁₁

Else {

set c = 0

for i = 1 to n,

matrix B = matrix A with row i and column 1 deleted.

c = c + (-1)^(i-1) a_i1 |B|

return c

}

Related Pages in this website

RREF -- Reduced Row Echelon Form

Cramer's Rule

Vectors -- the "dot" product and the "cross" product, explained.

The webmaster and author of the Math Help site is Graeme McRae.
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