EECM 3714
Lecture 12: Unit 12
Matrix algebra
Renshaw, Ch. 19
10 May 2022
,OUTLINE
Renshaw, Ch. 19
• Definitions, notation
• Matrix operations (Transposition – page 620; Addition; subtraction; Scalar multiplication; M
multiplication - Page 621-3)
• Determinants
• Matrix inversion
• 2 by 2 inversion
• 3 by 3 inversion
• Solving systems of linear equations (Matrix Inversion and Cramer’s rule)
,DEFINITIONS, NOTATION
• Matrix is a rectangular array of numbers/variables, e.g. (Page 578-579):
𝑎 𝑏 𝑐
• 𝐴3×3 = 𝑑 𝑒 𝑓 , 𝐵2×2 = 1 2
3 0
𝑔 ℎ 𝑖
• Order = dimensions of a matrix
• Order = number of rows (r) by number of columns (c)
• Usually denoted as m n, m = rows, n = columns
• An element is an entry in a matrix, denoted as 𝑎𝑖𝑗 , e.g. the element 𝑎23 = 𝑓 in matrix A,
the element 𝑏22 = 0 in matrix B
, SPECIAL MATRICES VECTORS AND SCALARS
• Square matrix: number of rows = number of • Scalar is a 1 × 1 matrix, i.e. a consta
columns, i.e. 𝑚 = 𝑛 • Row vector: matrix with only one
1 3 5 i.e. 𝑚 = 1, e.g. 𝑅 = 1 5 2
• E.g.𝐶 = 7 6 4
• Column vector = matrix with only
0 23 1 2
• Null matrix: every element of matrix = 0 e.g. column, i.e. 𝑛 = 1, e.g. 𝐷 = 4
0 0 1
0=
0 0
• Identity matrix: diagonal elements are all 1; all
other elements are 0
1 0
• Note: must be a square matrix, e.g. 𝐼 =
0 1
Lecture 12: Unit 12
Matrix algebra
Renshaw, Ch. 19
10 May 2022
,OUTLINE
Renshaw, Ch. 19
• Definitions, notation
• Matrix operations (Transposition – page 620; Addition; subtraction; Scalar multiplication; M
multiplication - Page 621-3)
• Determinants
• Matrix inversion
• 2 by 2 inversion
• 3 by 3 inversion
• Solving systems of linear equations (Matrix Inversion and Cramer’s rule)
,DEFINITIONS, NOTATION
• Matrix is a rectangular array of numbers/variables, e.g. (Page 578-579):
𝑎 𝑏 𝑐
• 𝐴3×3 = 𝑑 𝑒 𝑓 , 𝐵2×2 = 1 2
3 0
𝑔 ℎ 𝑖
• Order = dimensions of a matrix
• Order = number of rows (r) by number of columns (c)
• Usually denoted as m n, m = rows, n = columns
• An element is an entry in a matrix, denoted as 𝑎𝑖𝑗 , e.g. the element 𝑎23 = 𝑓 in matrix A,
the element 𝑏22 = 0 in matrix B
, SPECIAL MATRICES VECTORS AND SCALARS
• Square matrix: number of rows = number of • Scalar is a 1 × 1 matrix, i.e. a consta
columns, i.e. 𝑚 = 𝑛 • Row vector: matrix with only one
1 3 5 i.e. 𝑚 = 1, e.g. 𝑅 = 1 5 2
• E.g.𝐶 = 7 6 4
• Column vector = matrix with only
0 23 1 2
• Null matrix: every element of matrix = 0 e.g. column, i.e. 𝑛 = 1, e.g. 𝐷 = 4
0 0 1
0=
0 0
• Identity matrix: diagonal elements are all 1; all
other elements are 0
1 0
• Note: must be a square matrix, e.g. 𝐼 =
0 1
