The numbers inside a matrix are called the elements or entries of the matrix.

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1 Chapter Review of Matries. Definitions A matrix is a retangular array of numers of the form a a a 3 a n a a a 3 a n a 3 a 3 a 33 a 3n..... a m a m a m3 a mn We usually use apital letters (for example, A, B, C,...) for the names of matries, an we usually use lowerase letters (for example, a,,,...) to represent the numers insie of a matrix. The numers insie a matrix are alle the elements or entries of the matrix. The sequene of all entries on a horizontal line is alle a row, an the sequene of all entries on a vertial line is alle a olumn. We numer the rows from top to ottom, an the olumns from left to right. The entry in the kth row an lth olumn of a matrix A is enote y a kl. A matrix with m rows an n olumns is alle an m n matrix, or a matrix of orer m n. Example (a) A = is a 3 4 matrix sine it has 3 rows an 4 olumns. 3 4 The entry in the n row an 4th olumn of A is 3, an so we write a 4 = 3.

2 B = 3 4 is a 4 matrix. It an also e referre to as a olumn matrix or a olumn vetor. () C = 3 4 is a 5 matrix. It an also e referre to as a row matrix or row vetor. Further Definitions: If a matrix has orer n n (that is, if the numer of rows equals the numer of olumns) then the matrix is alle square. The main iagonal of a square matrix onsists of the entries on the iagonal from the top left orner of the matrix own to the ottom right orner of the matrix. A iagonal matrix is a square matrix in whih all of the entries whih are not on the main iagonal must equal zero. A triangular matrix is a square matrix in whih all the entries whih lie elow the main iagonal are zero, or all the entries whih lie aove the main iagonal are zero. Example... (a) D = E = is a square matrix. is a 3 3 square matrix. We see that E is a 3 3 iagonal matrix. (Note that E is also a 3 3 triangular matrix.) () F = an G = are 3 3 triangular matries.

3 . Matrix Operations Matrix equality. Two matries are equal if they have the same orer an their orresponing entries are equal. a = e g f h if an only if a = e, = f, = g an = h. Matrix aition. Two matries an e ae if they have the same size. We a two suh matries y aing the orresponing entries. a + e g f h = a + e + g + f + h For example, = Salar multipliation. Any matrix an e multiplie y a single numer (salar). We o this y multiplying all the entries of the matrix y that numer. k a = ka k k k For example, 4 3 = Matrix multipliation. The prout AB of two matries A an B is efine if an only if the numer of olumns of A is equal to the numer of rows of B. n olumns n rows A B To alulate the entry in the (i, j) position (that is, in the i th row an j th olumn) of AB, we multiply the i th row of A y the j th olumn of B. 3

4 Example... a e g f h = ae + g e + g af + h f + h If A has orer m n an B has orer n p then AB has orer m p Example a e f = 3a e 3 + f 4a e 4 + f Example = = Transpose. The transpose A T of a matrix A is otaine y putting the ith row of A into the ith olumn of A T. If A has orer m n then A T has orer n m. Example..4. If A = then A T =

5 .3 Properties of Matrix Operations.3. Aition Properties A + B = B + A A + (B + C) = (A + B) + C A + O = A = O + A A + A = O = A + A The matrix O is alle the zero matrix, an A is alle the negative of A. In the aove properties, A, B, C, O an A all have the same orer. Thus, for example, (a) if A = a then O = an A = a if A = a e f then O = an A = a e f.3. Multipliation Properties Provie that eah of the following matrix prouts exists, we have A(B + C) = AB + AC (B + C)A = BA + CA A(BC) = (AB)C However, usually AB BA For a square matrix A, we have: AO = O = OA an AI = A = IA where I is a iagonal matrix with ones on the main iagonal an zeros elsewhere. The matrix I is alle the ientity matrix, an must have the same orer as A. 5

6 For example: If A is then I = If A is 3 3 then I = Note: It is important to realize that, if we hoose any two matries A an B, then usually AB BA For example, onsier the matries A an B given elow: Example.3.. Let A = an B = Then AB = = = an BA = = = In this example, AB BA 6

7 .3.3 Exerises. Fin (a) Fin (a) () Fin (a) Given that fin x x = , 7

8 .4 The Determinant an Inverse of a Matrix. The matrix B is the inverse of A if AB = I = BA. Not all matries have inverses. When a matrix A oes have an inverse B, then the inverse is unique (i.e., it has no other inverses), an we write A instea of B. Thus AA = I = A A For A = a, we efine the eterminant of A y et(a) = a We sometimes write a instea of et(a). If et(a) 0 then we have A = et(a) a. Then A satisfies as require. AA = I = A A Note: If et(a) = 0 then A oes not exist, an we say that A is singular. Example.4.. Fin the eterminant an the inverse of the matrix A = 3. Solution. We have et(a) = 3 = = an A = 3 = =

9 .4. Exerises. Evaluate 0 (a) 0 () 3. Fin the inverse of the following matries: (a) () Solving Matrix Equations Consier the matrix equation AX = C, where X is an unknown matrix. We annot ivie y A. Instea, we multiply oth sies of the equation y A (if A exists). In partiular, if A exists then we an solve the aove equation for the matrix X, as follows: AX = C A AX = A C (multiplying oth sies of the equation on the left y A ) IX = A C X = A C. Similarly, XA = C XAA = CA (multiplying oth sies of the equation on the right y A ) X = CA. Note that Multiplying on the left is alle premultiplying. Multiplying on the right is alle postmultiplying. 9

10 Example.5.. Fin the matrix X suh that X 3 = 9 0. Solution. Sine Now 3 is on the right of X, we multiply on the right y X 3 = X 3 3 = XI = X = = = Uniqueness of Solutions Sometimes there an e more than one matrix X suh that AX = C, or there oul e none! When there is only one X suh that AX = C, we say that AX = C has a unique solution. Consier the matrix equation AX = C. We have the following result: When et(a) 0 then AX = C has a unique solution. This unique solution is given y X = A C. When et(a) = 0 then AX = C either has infinitely many solutions, or no solutions. 0

11 .5. Exerises. Fin the matrix X suh that (a) X 9 = X 5 = () X 0 3 = 0 () X 0 0 = 0 0. Fin the matrix Y suh that (a) 3 Y = Y = () 5 Y = () 5 Y = 3 3. Fin the matrix Z suh that (a) 3 Z 5 = 4 0 Z 0 4 = Simultaneous Equations with unknowns An equation of the form ax + y = p (where a, an p are onstants) represents a straight line in the x, y-plane. The system of equations ax + y = p x + y = q represents two lines in the x, y-plane. When we solve these equations simultaneously, we are fining the point of intersetion of the two lines. We an write the aove system of equations in matrix form as follows: a x = p. y q

12 Example.6.. Solve the following system of equations: x + 3y = 8 x 4y = 7 Solution. We nee to fin x an y suh that oth of the aove equations are satisfie. To o this, we write the aove system in matrix form: 3 4 x = y Then we premultiply the matrix equation y the inverse of to get x y = = = =. Thus x = an y =.

13 .6. To Solve a System of Equations: (a) Write the system of equations ax + y = p x + y = q in matrix form: a x y = p q. Calulate the eterminant = a = a. () If 0 then we have a unique solution given y x = a p y q. () If = 0 then either there are no solutions. In this ase, the equations represent two parallel lines whih on t interset. or there are infinitely many solutions. In this ase the equations represent the same line, an so any point on this line is a solution. 3

14 Example.6.. Solve the following system of equations: x + 3y = 6 4x + 6y = Solution. In matrix form, we have x y = 6 Note that the eterminant 3 = = = Therefore, there is no solution or there are infinitely many solutions. Graphially, the lines x+3y = 6 an 4x+6y = are non-interseting parallel lines: y x + 3y = x 4x + 6y = Therefore, there is no solution. 4

15 Example.6.3. is a onstant: Consier the following simultaneous system of linear equations, where k kx + 3y = 4 3x + ky = 5. (a) Write own the system of equations in matrix form AX = B. Solution. k 3 3 k x y = 4 5. Calulate the eterminant of the matrix A foun in (a). Solution. A = k 3 3 k an so et(a) = k 3 3 k = k 9. () Fin the value(s) of k for whih the system has a unique solution. Solution. There is a unique solution if an only if et(a) 0 k 9 0 (k 3)(k + 3) 0 k 3 an k 3 k R \ {3, 3}. 5

16 .6. Exerises. Solve the following systems of equations: (a) x + 3y = 8 x + 4y = 9 x y = 5 x + 3y = () x + 3y = 6 4x + 6y = 3 () x + 3y = 6 4x + 6y =. Consier the following simultaneous system of linear equations, where p is a onstant: 3x + py = 9 (p + )x + y = 9. (a) Show that the system has a unique solution if an only if p R \ { 3, }. Fin the value(s) of p for whih the system has infinitely many solutions. () Fin the value(s) of p for whih the system has no solution. Hint for an (): Consier the ases p = 3 an p = separately, i.e., sustitute p = 3 into the system an then try to solve the system; o the same for p =. 6

17 Chapter 5 Answers 5. Answers for Chapter Exerises.3.3:. (a) (a) 3. (a) () 8 4. x = 9.4.:. (a) 0 () 0 0. (a) () :. (a). (a) 3. (a) () () 0 3 () () :. (a) (, ) (, ) () No solution () More than one solution: any point on the line x + 3y = 6.. (a) Omitte p = () p = 3. 59