Determinants - Solved Examples – 2×2 matrix

• A determinant is a scalar value that can be calculated from the elements of a square matrix.
• The determinant of a 2×2 matrix [ a b ; c d ] is calculated as ad - bc. Example: Given the matrix [ 3 4 ; 2 1 ], we can calculate its determinant as follows:
• Determinant = (3 * 1) - (4 * 2)
• Determinant = 3 - 8
• Determinant = -5 Equation: Determinant = ad - bc

Determinants - Solved Examples – 3×3 matrix

• For a 3×3 matrix [ a b c ; d e f ; g h i ], the determinant can be calculated using the formula:
  • Determinant = a(ei - fh) - b(di - fg) + c(dh - eg). Example: Given the matrix [ 1 0 2 ; -1 3 1 ; 4 -2 0 ], we can calculate its determinant as follows:
• Determinant = 1(3 * 0 - 1 * (-2)) - 0((-1) * 0 - 4 * (-2)) + 2((-1) * (-2) - 4 * 3)
• Determinant = 1(0 + 2) - 0(0 + 8) + 2(2 - 12)
• Determinant = 2 - 0 + (-20)
• Determinant = -18 Equation: Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)

Matrices - Addition

• In matrices, addition is performed by adding corresponding elements of two matrices.
• Two matrices can only be added or subtracted if they have the same dimensions. Example: Given two matrices A and B:
• A = [ 2 4 ; 1 3 ]
• B = [ 5 1 ; 2 6 ] We can add them as follows:
• A + B = [ 2+5 4+1 ; 1+2 3+6 ]
• A + B = [ 7 5 ; 3 9 ] Equation: (A + B)ij = Aij + Bij

Matrices - Subtraction

• Subtraction of two matrices is similar to addition, but with opposite signs for the elements of the second matrix.
• Just like addition, two matrices can be subtracted only if they have the same dimensions. Example: Given two matrices A and B:
• A = [ 2 4 ; 1 3 ]
• B = [ 5 1 ; 2 6 ] We can subtract them as follows:
• A - B = [ 2-5 4-1 ; 1-2 3-6 ]
• A - B = [ -3 3 ; -1 -3 ] Equation: (A - B)ij = Aij - Bij

Matrices - Scalar Multiplication

• Scalar multiplication involves multiplying each element of a matrix by a scalar (a real number).
• The resulting matrix has the same dimensions as the original matrix. Example: Given a matrix A = [ 2 4 ; 1 3 ] and a scalar k = 3, we can perform scalar multiplication as follows:
• kA = [ 3 * 2 3 * 4 ; 3 * 1 3 * 3 ]
• kA = [ 6 12 ; 3 9 ] Equation: (kA)ij = k * Aij

Matrices - Matrix Multiplication

• Matrix multiplication involves multiplying the elements of one matrix with the corresponding elements of another matrix.
• The resulting matrix is obtained by summing the products of corresponding elements from the two matrices. Example: Given two matrices A and B:
• A = [ 2 3 ; 1 -2 ]
• B = [ -5 4 ; 3 6 ] We can multiply them as follows:
• AB = [ (2 * -5) + (3 * 3) (2 * 4) + (3 * 6) ; (1 * -5) + (-2 * 3) (1 * 4) + (-2 * 6) ]
• AB = [ -1 30 ; -11 -8 ] Equation: (AB)ij = ∑ (Aik * Bkj), for k from 1 to n

Matrices - Transpose

• The transpose of a matrix is obtained by interchanging its rows with columns.
• The resulting matrix has dimensions opposite to the original matrix. Example: Given a matrix A = [ 2 4 ; 1 3 ], we can find its transpose as follows:
• AT = [ 2 1 ; 4 3 ] Equation: (AT)ij = Aji

Matrices - Identity Matrix

• An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere.
• Multiplying any matrix by an identity matrix gives the same matrix. Example: The 2×2 identity matrix is given by:
• I = [ 1 0 ; 0 1 ] Equation: IM = MI = M, for any matrix M

Matrices - Inverse

• The inverse of a matrix is denoted as A^-1 and is defined as:
  • A * A^-1 = A^-1 * A = I, where I is the identity matrix.
• A matrix is invertible if its determinant is non-zero. Example: Given a matrix A = [ 2 3 ; 1 -2 ], we can find its inverse as follows: Step 1: Calculate the determinant of A.
• Determinant = (2 * -2) - (3 * 1)
• Determinant = -4 - 3
• Determinant = -7 Step 2: Find the adjoint of A.
• Adjoint = [ -2 -3 ; -1 2 ] Step 3: Calculate the inverse of A.
• A^-1 = (1 / Determinant) * Adjoint
• A^-1 = (1 / -7) * [ -2 -3 ; -1 2 ]
• A^-1 = [ 2/7 3/7 ; 1/7 -2/7 ] Equation: A * A^-1 = A^-1 * A = I

Determinants - Solved Examples – 2×2 matrix

• Example 1: Given the matrix [ 3 4 ; 2 1 ], we can calculate its determinant as follows:
  • Determinant = (3 * 1) - (4 * 2)
  • Determinant = 3 - 8
  • Determinant = -5
  • Equation: Determinant = ad - bc
• Example 2: Given the matrix [ 5 -2 ; 3 6 ], we can calculate its determinant as follows:
  • Determinant = (5 * 6) - (-2 * 3)
  • Determinant = 30 + 6
  • Determinant = 36
  • Equation: Determinant = ad - bc
• Example 3: Given the matrix [ -3 0 ; 1 -4 ], we can calculate its determinant as follows:
  • Determinant = (-3 * -4) - (0 * 1)
  • Determinant = 12 - 0
  • Determinant = 12
  • Equation: Determinant = ad - bc
• Example 4: Given the matrix [ 2 3 ; -1 -2 ], we can calculate its determinant as follows:
  • Determinant = (2 * -2) - (3 * -1)
  • Determinant = -4 + 3
  • Determinant = -1
  • Equation: Determinant = ad - bc
• Example 5: Given the matrix [ 0 -1 ; 2 4 ], we can calculate its determinant as follows:
  • Determinant = (0 * 4) - (-1 * 2)
  • Determinant = 0 + 2
  • Determinant = 2

  • Equation: Determinant = ad - bc

Determinants - Solved Examples – 3×3 matrix

• Example 1: Given the matrix [ 1 0 2 ; -1 3 1 ; 4 -2 0 ], we can calculate its determinant as follows:
  • Determinant = 1(3 * 0 - 1 * (-2)) - 0((-1) * 0 - 4 * (-2)) + 2((-1) * (-2) - 4 * 3)
  • Determinant = 1(0 + 2) - 0(0 + 8) + 2(2 - 12)
  • Determinant = 2 - 0 + (-20)
  • Determinant = -18
  • Equation: Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)
• Example 2: Given the matrix [ 5 1 -3 ; 2 -1 4 ; 6 3 0 ], we can calculate its determinant as follows:
  • Determinant = 5((-1 * 0) - (4 * 3)) - 1((2 * 0) - (6 * 3)) + (-3)((2 * 3) - (6 * -1))
  • Determinant = 5(0 - 12) - 1(0 - 18) + (-3)(6 - (-6))
  • Determinant = 5(-12) - 1(-18) + (-3)(12)
  • Determinant = -60 + 18 - 36
  • Determinant = -78
  • Equation: Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)
• Example 3: Given the matrix [ 3 -4 2 ; 1 5 3 ; -2 0 -1 ], we can calculate its determinant as follows:
  • Determinant = 3((5 * -1) - (3 * 0)) - (-4)((1 * -1) - (-2 * 0)) + 2((1 * 0) - (-2 * 5))
  • Determinant = 3(5 - 0) - (-4)(1 - 0) + 2(0 + 10)
  • Determinant = 3(5) - (-4)(1) + 2(10)
  • Determinant = 15 + 4 + 20
  • Determinant = 39
  • Equation: Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)
• Example 4: Given the matrix [ 2 1 3 ; 0 4 -2 ; -1 0 1 ], we can calculate its determinant as follows:
  • Determinant = 2((4 * 1) - (-2 * 0)) - 1((0 * 1) - (-1 * -2)) + 3((0 * 0) - (-1 * 4))
  • Determinant = 2(4) - 1(0 - 2) + 3(0 - (-4))
  • Determinant = 8 - 1(2) + 3(4)
  • Determinant = 8 - 2 + 12
  • Determinant = 18
  • Equation: Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)
• Example 5: Given the matrix [ 0 -2 1 ; 3 0 -4 ; -1 -3 2 ], we can calculate its determinant as follows:
  • Determinant = 0((0 * 2) - (-4 * -3)) - (-2)((3 * 2) - (-1 * -4)) + 1((3 * -3) - (-1 * 0))
  • Determinant = 0(0 - 12) - (-2)(6 - 4) + 1(-9 - 0)
  • Determinant = 0 - (-2)(2) - 1(-9)
  • Determinant = 0 + 4 + 9
  • Determinant = 13

  • Equation: Determinant = a(ei - fh) - b(di - fg) + c(dh - eg)

Matrices - Addition

• Addition of two matrices is performed by adding corresponding elements of the matrices.
• Two matrices can only be added if they have the same dimensions.
• Example:
  • Given two matrices A and B:
    • A = [ 2 4 ; 1 3 ]
    • B = [ 5 1 ; 2 6 ]
  • We can add them as follows:
    • A + B = [ 2+5 4+1 ; 1+2 3+6 ]
    • A + B = [ 7 5 ; 3 9 ]
  • Equation: (A + B)ij = Aij + Bij
• Additional Example:
  • Given two matrices C and D:
    • C = [ 7 3 ; 5 -2 ]
    • D = [ -1 2 ; 4 0 ]
  • We can add them as follows:
    • C + D = [ 7+(-1) 3+2 ; 5+4 -2+0 ]
    • C + D = [ 6 5 ; 9 -2 ]
  • Equation: (C + D)ij = Cij + Dij
• Additional Example:
  • Given two matrices E and F:
    • E = [ 1 -2 ; 0 3 ]
    • F = [ 4 1 ; -3 2 ]
  • We can add them as follows:
    • E + F = [ 1+4 -2+1 ; 0+(-3) 3+2 ]
    • E + F = [ 5 -1 ; -3 5 ]
  • Equation: (E + F)ij = Eij + Fij
• Additional Example:
  • Given two matrices G and H:
    • G = [ 0 0 ; 0 0 ]
    • H = [ 1 1 ; 1 1 ]
  • We can add them as follows:
    • G + H = [ 0+1 0+1 ; 0+1 0+1 ]
    • G + H = [ 1 1 ; 1 1 ]
  • Equation: (G + H)ij = Gij + Hij
• Additional Example:
  • Given two matrices I and J:
    • I = [ -1 2 4 ; 0 3 -2 ; 1 -1 -3 ]
    • J = [ 3 -2 1 ; -4 0 3 ; -1 2 -1 ]
  • We can add them as follows:
    • I + J = [ -1+3 2+(-2) 4+1 ; 0+(-4) 3+0 -2+2 ; 1+(-1) -1+2 -3+(-1) ]
    • I + J = [ 2 0 5 ; -4 3 0 ; 0 1 -4 ]

  • Equation: (I + J)ij = Iij + Jij

Matrices - Subtraction

• Subtraction of two matrices is similar to addition, but with opposite signs for the elements of the second matrix.
• Two matrices can be subtracted only if they have the same dimensions.
• Example:
  • Given two matrices A and B:
    • A = [ 2 4 ; 1 3 ]
    • B = [ 5 1 ; 2 6 ]
  • We can subtract them as follows:
    • A - B = [ 2-5 4-1 ; 1-2 3-6 ]
    • A - B = [ -3 3 ; -1 -3 ]
  • Equation: (A - B)ij = Aij - Bij
• Additional Example:
  • Given two matrices C and D:
    • C = [ 7 3 ; 5 -2 ]
    • D = [ -1 2 ; 4 0 ]
  • We can subtract them as follows:
    • C - D = [ 7-(-1) 3-2 ; 5-4 -2-0 ]
    • C - D = [ 8 1 ; 1 -2 ]
  • Equation: (C - D)ij = Cij - Dij
• Additional Example:
  • Given two matrices E and F:
    • E = [ 1 -2 ; 0 3 ]
    • F = [ 4 1 ; -3 2 ]
  • We can subtract them as follows:
    • E - F = [ 1-4 -2-1 ; 0-(-3) 3-2 ]
    • E - F = [ -3 -3 ; 3 1 ]
  • Equation: (E - F)ij = Eij - Fij
• Additional Example:
  • Given two matrices G and H:
    • G = [ 0 0 ; 0 0 ]
    • H = [ 1 1 ; 1 1 ]
  • We can subtract them as follows:
    • G - H = [ 0-1 0-1 ; 0-1 0-1 ]
    • G - H = [ -1 -1 ; -1 -1 ]
  • Equation: (G - H)ij = Gij - Hij
• Additional Example:
  • Given two matrices I and J:
    • I = [ -1 2 4 ; 0 3 -2 ; 1 -1 -3 ]
    • J = [ 3 -2 1 ; -4 0 3 ; -1 2 -1 ]
  • We can subtract them as follows:
    • I - J = [ -1-3