MATRICES & DETERMINANTS Monika V Sikand Light and Life Laboratory Department of Physics and Engineering physics Stevens Institute of Technology Hoboken, New Jersey, 07030.
OUTLINE • Matrix Operations • Multiplying Matrices • Determinants and Cramer’s Rule • Identity and Inverse Matrices • Solving systems using Inverse matrices
MATRIX A rectangular arrangement of numbers in rows and columns For example: 2 rows 3 columns
COMPARING MATRICES EQUAL MATRICES: Matrices having equal corresponding entries. For Example:
ADDING MATRICES Matrices of same dimension can be added For Example:
SUBTRACTING MATRICES Matrices of same dimension can be subtracted For example:
MULTIPLYING A MATRIX BY A SCALAR For example:
SOLVING A MATRIX EQUATION For example:
PRODUCT OF TWO MATRICES For example: FIND (a.) AB and (b.) BA
SIMPLIFY Simplify: a.) A(B+C) b.) AB+AC
DETERMINANT OF 22 MATRIX The determinant of a 22 matrix is the difference of the entries on the diagonal.
EVALUATE Find the determinant of the matrix: Solution:
DETERMINANT OF 33 MATRIX The determinant of a 33 matrix is the difference in the sum of the products in red from the sum of theproducts in black. Determinant = [a(ei)+b(fg)+c(dh)]-[g(ec)+h(fa)+i(db)]
USING MATRICES IN REAL LIFE The Bermuda Triangle is a large trianglular region in the Atlantic ocean. Many ships and airplanes have been lost in this region. The triangle is formed by imaginary lines connecting Bermuda, Puerto Rico, and Miami, Florida. Use a determinant to estimate the area of the Bermuda Triangle. N Bermuda (938,454) . . Miami (0,0) W E . Puerto Rico (900,-518) S
SOLUTION The approximate coordinates of the Bermuda Triangle’s three vertices are: (938,454), (900,-518), and (0,0). So the area of the region is as follows: Hence, area of the Bermuda Triangle is about 447,000 square miles.
USING MATRICES IN REAL LIFE The Golden Triangle is a large triangular region in the India.The Taj Mahal is one of the many wonders that lie within the boundaries of this triangle. The triangle is formed by the imaginary lines that connect the cities of New Delhi, Jaipur, and Agra. Use a determinant to estimate the area of the Golden Triangle. The coordinates given are measured in miles. . N New Delhi (100,120) . . Jaipur (0,0) W E Agra (140,20) S
SOLUTION The approximate coordinates of the Golden Triangle’s three vertices are: (100,120), (140,20), and (0,0). So the area of the region is as follows: Hence, area of the Golden Triangle is about 7400 square miles.
USING MATRICES IN REAL LIFE Black neck stilts are birds that live throughout Florida and surrounding areas but breed mostly in the triangular region shown on the map. Use a determinant to estimate the area of this region. The coordinates given are measured in miles. . N (35,220) . . (112,56) W E (0,0) S
SOLUTION The approximate coordinates of the Golden Triangle’s three vertices are: (35,220), (112,56), and (0,0). So the area of the region is as follows: Hence, area of the region is about 11340 square miles.
CRAMER”S RULE FOR A 22 SYSTEM Let A be the co-efficient matrix of the linear system: ax+by= e & cx+dy= f. IF det A ≠0, then the system has exactly one solution. The solution is: The numerators for x and y are the determinant of the matrices formed by using the column of constants as replacements for the coefficients of x and y, respectively.
EXAMPLE Use cramer’s rule to solve this system: 8x+5y = 2 2x-4y = -10
SOLUTION Solution: Evaluate the determinant of the coefficient matrix Apply cramer’s rule since the determinant is not zero. The solution is (-1,2)
CRAMER”S RULE FOR A 33 SYSTEM Let A be the co-efficient matrix of the linear system: ax+by+cz= j, dx+ey+fz= k, and gx+hy+iz=l. IF det A ≠0, then the system has exactly one solution. The solution is:
EXAMPLE The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon(C ), hydrogen(H), and oxygen(O).
SOLUTION Write a linear system using the formula for each compound C + 4H = 16 3C+ 8H + 3O = 92 2H + O =18 Evaluate the determinant of the coefficient matrix.
SOLUTION Apply cramer’s rule since determinant is not zero. Atomic weight of carbon = 12 Atomic weight of hydrogen =1 Atomic weight of oxygen =16
IDENTITY MATIX 22 IDENTITY MATRIX 33 IDENTITY MATRIX
INVERSE MATRIX The inverse of the matrix
EXAMPLE Find the inverse of Solution:
SOLVING A LINEAR SYSTEM -3x + 4y = 5 2x - y = -10 Writing the original matrix equation. A X B AX = B A-1AX = A-1B IX = A-1B X = A-1B
USING INVERSE MATRIX TO SOLVE THE LINEAR SYSTEM -3x + 4y = 5 2x - y = -10 Hence the solution of the system is (-7,-4)