同济大学：《线性代数》课程教学资源（课件讲稿）Determinants

Determinants Li Zhang Department of Mathematics Tongji University 4907 Tongji University】 1/32

. . Determinants Li Zhang Department of Mathematics Tongji University (Tongji University) Linear Algebra 1 / 32

Introduction 1.Linear Equation: a1x1+·…+amxn=b. 2.a simple systems of linear equations: a1十12x十-十41nw=b 2x1+x2+十y=b dm1十d2十十du=bn Tongji University】 2/32

Introduction 1. Linear Equation: a1x1 + · · · + anxn = b. 2. a simple systems of linear equations：    a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 . . . . . . am1x1 + am2x2 + · · · + amnxn = bm (Tongji University) Linear Algebra 2 / 32

Introduction 1.Linear Equation: a1x1+·…+amxn=b. 2.a simple systems of linear equations: a11x1+a12x2+··+a41mxm=b1 a21x1+a22x2+··+a2n-xn=b2 am1x1+am2X2十··+AmnXn=bm Tongji University】 2/32

Introduction 1. Linear Equation: a1x1 + · · · + anxn = b. 2. a simple systems of linear equations：    a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 . . . . . . am1x1 + am2x2 + · · · + amnxn = bm (Tongji University) Linear Algebra 2 / 32

Introduction 1.Linear Equation: a1x1+·…+amxn=b. 2.a simple systems of linear equations: a11x1+a12x2+··+a41mxm=b1 a21x1+a22x2+··+a2n-xn=b2 am1x1+am2X2十··+AmnXn=bm Tongji University】 2/32

Introduction 1. Linear Equation: a1x1 + · · · + anxn = b. 2. a simple systems of linear equations：    a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 . . . . . . am1x1 + am2x2 + · · · + amnxn = bm (Tongji University) Linear Algebra 2 / 32

Matrix Definition 7.1.1 a11... 41n .. am1 ..。amn Note:m x n matrix,square matrix,entry,order,main diagonal,row matrix, column matrix (Tongji University) 3/32

Matrix . Definition 7.1.1 . .   a11 . . . a1n . . . . . . am1 . . . amn   Note: m × n matrix, square matrix, entry, order, main diagonal, row matrix, column matrix (Tongji University) Linear Algebra 3 / 32

Matrix Definition 7.1.1 a11... 41n .. aml ..。amn Note:m x n matrix,square matrix,entry,order,main diagonal,row matrix, column matrix (Tongji University) 3/32

Matrix . Definition 7.1.1 . .   a11 . . . a1n . . . . . . am1 . . . amn   Note: m × n matrix, square matrix, entry, order, main diagonal, row matrix, column matrix (Tongji University) Linear Algebra 3 / 32

Determinant ●2×2—Diagonal1 ethod 1D20a 411a12 Secondary Diae do aa2 (Tongji University) 4/32

Determinant 2 × 2—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag (Tongji University) Linear Algebra 4 / 32

Determinant ●2×2-Diagonal Method Main Diagonala1 21 =110 Secondary Diag 212 Tongji University】 4/32

Determinant 2 × 2—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag (Tongji University) Linear Algebra 4 / 32

Determinant e2×2-Diagonal Method Main Diagonal a11 a12 =111-0021 Secondary Diag a21 a22 Tongji University】 4/32

Determinant 2 × 2—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag (Tongji University) Linear Algebra 4 / 32

Determinant e2×2-Diagonal Method Main Diagonal a11 a12 =a11a22 -012021 Secondary Diag a21 a22 (Tongji University) 4/32

Determinant 2 × 2—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag (Tongji University) Linear Algebra 4 / 32