Mathematics Cheat Sheet for Population Biology - Stanford University

⎡n(t) = ⎣⎤n 1n 2⎦n 3A scalar is a single number: λ = 1.05We refer to individual matrix elements by indexing them by their row and column positions.A matrix is typically named by a capital (bold) letter (e.g., A). An element of matrix A isgiven by a lowercase a subscripted with its indices. These indices are subscripted following thethe lowercase letter, first by row, then by column. For example, a 21 is the element of A whichis in the second row and first column.Matrix Multiplication[a11 a 12a 21a 22] [n1n 2]=Multiply each row element-wise by the columnFor Example,[ 2 34 5] [ 67]=[ ]a11 n 1 + a 12 n 2a 21 n 1 + a 22 n 2[ (2 · 6) + (3 · 7)(4 · 6) + (5 · 7)]=[ 3359Matrix Addition or Subtraction[ ] [ ] [ ]a11 a 12 b11 b+12 a11 + b=11 a 12 + b 12a 21 a 22 b 21 b 22 a 21 + b 21 a 22 + b 22[ 1 23 4]+[ 5 67 8]=[ 6 810 12Multiplying a Matrix by a Scalar[ ] [ ]a11 aλ12 λa11 λa=12a 21 a 22 λa 21 λa 224[ 2 34 5]=[ 8 1216 20]]]Systems of Equationseasier.Matrix notation was invented to make solving simultaneous equationsy 1 = ax 1 + bx 2In matrix notation:y 2 = cx 1 + dx 2[ ] [y1 a b=y 2 c d] [x1x 2]7