Let U = {A E M2] det(A) = 0} where M2 is the vector space of...

              

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Let U = {A E M₂2] det(A) = 0} where M₂2 is the vector space of two by two matrices. Find an example that shows U is NOT a subspace of M22. In cryptography we often use matrix operations but modulo some M for the calculations (for example, in Hill substitution ciphers.) Do the following mod 26 calculations (see the attached multiplication table mod 26): a. Compute the following matrix product modulo 26 [17 22 4 [21 1 18] 2 9 11 25 22 20 4 6 13] [17 24 19] b. A "key matrix" is use to encode a message where blocks of letters are replaced by blocks of numbers (for example, BAD might become 204.) To decrypt the message, it is necessary to invert the key matrix (mod 26). If the key matrix is given as, [10 18 1 1 3 7 21 A = 17 4 then Compute A-1 mod 26. (Hint: Row reduce the augmented matrix using modulo 26 calculations. Recall that the modulus of a number N is the reminder after dividing by the modulus M. For example, 55 is equal to (we say "congruent to") 3 modulo (or "mod 26") since 55 = 2(26) + 3.) You may use software to assist doing this. Multiplication table Modulo 26 BBARBಹ ವ ಹ ಹ ಹ ವನ-5 0 0 -Jan 7 A J N 42 11 13 20 22 0 23 ೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦ 24 0 0 12 BSRR8 ಹ ಹ ಸ ಹ ಗ ಹ ದ ನ - 5 0 0 -J ಾ – A W N - D 5 8 10 20 11 22 ಕ ೦88 ಕ a cla 8 = o n v KR ಹ ಹ ದ ಹ J_Y ಹ ಗ ನ ಾ ಟ – ಎ RB ಹ ಹ ದ ನ ಹan AvoRRB ಹ ಹ ಹ ಸ ಹ ಯ ಾ A N DIN ಹ ಹ ಹ P 8 ನ೦೦೦ ಹ ಹ ಹ ಹನ0A - A ಹ = - R ನ ನ Jಟ ಹ ದ ಯ 0 13 12 24 0 15 19 0 19 16 24 2 4 0 22 0 13 0 18 10 16 6 22 21 0 21 16 17 8 25 15 20 25 18 24 21 12 0 20 14 0 23 20 17 14 24 2 20 0 24 22 20 16 20 24 5 24 ಹನ-|| 8 23 6 6 ಕ ಕ 8 ಹನ-8 ಕ 05-8 10 22 8 20 4 14 22 16 6 10 16 22 4 12 20 14 25 7 24 7 8 ನ ೪ ನರ ಎ ನ ಗ ೦ - 8 ದಟ ಹ = 8 ಹRT J೦ 12 1 2 22 16 16 21 24 2 11 20 3 ಹ ಹ ದ 0 8 8 14 4 22 14 ಪ ನನ a & ಹ - ೪ ಪಟ ಕ ಟ ದ ನ 8- ಕ - ಹaa|a| ನ ನ ಹ ೦೪ ಪ8ಕ೦ಕಟ ನಹ ಉ೪ ಪ೪ ಕ೦ಕ > ಪ » & ನ - ಕ a Badna Rಕಟಕ ಹRo= adodad೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ ನ ಹ - ಹ ಾ & a Rಕನ೦ಪಕ - ಹ೦ಕಟ ಹ - ಹಣಕ೦ನRR8 ಹ ಹ ಹ ಹ 3 22 4 1 22 17 12 8 24 1 6 14 19 24 2 24 20 18 20 11 2 12 22 10 6 18 0 8 6 23 22 10 0 10 15 24 4 12 10 8 2 6 24 11 12 18 25 22 24 2 25 22 12 18 22 10 14 20 16 5 20 9 24 13 8 2 23 18 6 13 20 16 10 24 12 14 13 22 2 13 24 13 R - ಹ ದ ಕ ಟ ಕ ಟವ ವ 8 – ಹ - ನಟ ಹ ಹ ಹ - ಪ P ಕಟ8 5 22 13 23 18 13 10 20 4 14 24 16 17 6 8 19 22 25 8 12 16 16 20 24 2 18 23 2 12 17 22 1 6 18 24 16 22 10 8 15 22 16 24 6 14 22 4 14 23 15 24 18 9 20 5 12 24 10 22 16 17 18 19 20 21 21 22 22 23 23 24 25 0 0 0 20 14 4 0 22 2 8 0 18 10 2 20 1 20 13 6 25 18 11 6 0 20 14 8 3 24 19 14 10 ನ 8 ಹ೦೦ ಹRT RAನಕ ಹa|ಹ 6 22 12 4 21 12 3 20 16 13 10 7 4 6 4 2 0 24 22 20 18 # 80 ಕ ಟ - 1 ಹ ಜವ 8 - 0 do nನ ಹ - ಹ 8 20 13 0 13 16 4 6 21 10 25 14 3 12 23 6 0 13 0 TR 2 22 1 4 23 ಹPn 1 24 ಕd88೦೦ಕ ಹR ಎ ಉಪ ಹಟ ನನ 18 - Po/p 21 22 22 23 23 24 25 2 24 8 24 2 22 16 10 o ನ ಗ ಹ ೪ ೪ - - - ಹಪ ಹ ಹR & 70 + T 88-ಟ o ನ 88 ನರ ಸ ಹ ೦೭ ೦ನ ಹR ನ ಹ ಹ ಹRR 17 24 19 12 20 9 4 25 24 20 18 12 14 18 20 22 24 10 2 20 8 22 12 22 10 8 17 20 23 0 13 16 25 8 ಉ ರ ನ ನ ಹ ಹR೦ನದ ರಾ0 ರ ನ ಹ ಹ ಹRRog 24 20 16 2 10 18 22 12 6 24 14 20 10 24 7 22 4 20 1 18 2 21 14 V @ A I n -J 0 0 ರ ನ ದ ಹ ದ ಕ ವ ಹ ಹ8RR&R…! 2 14 12 8 6 > 4 21 19 18 16 15 11 10 6 2 4. [12] Use Gaussian elimination to determine the determinant of the matrix: 1 1 4 4 2 -2 8 -8- M = 1 1 1 -1 1 1 -1 -1 You may use software to do the reduction. is linearly independent. 5. [12] Consider two distinct numbers a and b and define the function - 1 1 1 b t b² t² f(t) = det a Answer the following: a) Show that f(t) is a quadratic function. What is the coefficient of t²? b) For what values of t is the matrix invertible? 6. [12] Use the determinant to find the values of the scalar λ for which the matrix A A3 fails to be invertible where, ГО A = 4 L3 2 6 2 305 7. [10] Find the value(s) of k for which the following set of vectors v₁ = (1, 0, 0) v₂ = (k, 1,k) V3 = (1, -2k, 3k + 1) Let U = {A E M₂2] det(A) = 0} where M₂2 is the vector space of two by two matrices. Find an example that shows U is NOT a subspace of M22. In cryptography we often use matrix operations but modulo some M for the calculations (for example, in Hill substitution ciphers.) Do the following mod 26 calculations (see the attached multiplication table mod 26): a. Compute the following matrix product modulo 26 [17 22 4 [21 1 18] 2 9 11 25 22 20 4 6 13] [17 24 19] b. A "key matrix" is use to encode a message where blocks of letters are replaced by blocks of numbers (for example, BAD might become 204.) To decrypt the message, it is necessary to invert the key matrix (mod 26). If the key matrix is given as, [10 18 1 1 3 7 21 A = 17 4 then Compute A-1 mod 26. (Hint: Row reduce the augmented matrix using modulo 26 calculations. Recall that the modulus of a number N is the reminder after dividing by the modulus M. For example, 55 is equal to (we say "congruent to") 3 modulo (or "mod 26") since 55 = 2(26) + 3.) You may use software to assist doing this. Multiplication table Modulo 26 BBARBಹ ವ ಹ ಹ ಹ ವನ-5 0 0 -Jan 7 A J N 42 11 13 20 22 0 23 ೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦೦ 24 0 0 12 BSRR8 ಹ ಹ ಸ ಹ ಗ ಹ ದ ನ - 5 0 0 -J ಾ – A W N - D 5 8 10 20 11 22 ಕ ೦88 ಕ a cla 8 = o n v KR ಹ ಹ ದ ಹ J_Y ಹ ಗ ನ ಾ ಟ – ಎ RB ಹ ಹ ದ ನ ಹan AvoRRB ಹ ಹ ಹ ಸ ಹ ಯ ಾ A N DIN ಹ ಹ ಹ P 8 ನ೦೦೦ ಹ ಹ ಹ ಹನ0A - A ಹ = - R ನ ನ Jಟ ಹ ದ ಯ 0 13 12 24 0 15 19 0 19 16 24 2 4 0 22 0 13 0 18 10 16 6 22 21 0 21 16 17 8 25 15 20 25 18 24 21 12 0 20 14 0 23 20 17 14 24 2 20 0 24 22 20 16 20 24 5 24 ಹನ-|| 8 23 6 6 ಕ ಕ 8 ಹನ-8 ಕ 05-8 10 22 8 20 4 14 22 16 6 10 16 22 4 12 20 14 25 7 24 7 8 ನ ೪ ನರ ಎ ನ ಗ ೦ - 8 ದಟ ಹ = 8 ಹRT J೦ 12 1 2 22 16 16 21 24 2 11 20 3 ಹ ಹ ದ 0 8 8 14 4 22 14 ಪ ನನ a & ಹ - ೪ ಪಟ ಕ ಟ ದ ನ 8- ಕ - ಹaa|a| ನ ನ ಹ ೦೪ ಪ8ಕ೦ಕಟ ನಹ ಉ೪ ಪ೪ ಕ೦ಕ > ಪ » & ನ - ಕ a Badna Rಕಟಕ ಹRo= adodad೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ೦ಪ ನ ಹ - ಹ ಾ & a Rಕನ೦ಪಕ - ಹ೦ಕಟ ಹ - ಹಣಕ೦ನRR8 ಹ ಹ ಹ ಹ 3 22 4 1 22 17 12 8 24 1 6 14 19 24 2 24 20 18 20 11 2 12 22 10 6 18 0 8 6 23 22 10 0 10 15 24 4 12 10 8 2 6 24 11 12 18 25 22 24 2 25 22 12 18 22 10 14 20 16 5 20 9 24 13 8 2 23 18 6 13 20 16 10 24 12 14 13 22 2 13 24 13 R - ಹ ದ ಕ ಟ ಕ ಟವ ವ 8 – ಹ - ನಟ ಹ ಹ ಹ - ಪ P ಕಟ8 5 22 13 23 18 13 10 20 4 14 24 16 17 6 8 19 22 25 8 12 16 16 20 24 2 18 23 2 12 17 22 1 6 18 24 16 22 10 8 15 22 16 24 6 14 22 4 14 23 15 24 18 9 20 5 12 24 10 22 16 17 18 19 20 21 21 22 22 23 23 24 25 0 0 0 20 14 4 0 22 2 8 0 18 10 2 20 1 20 13 6 25 18 11 6 0 20 14 8 3 24 19 14 10 ನ 8 ಹ೦೦ ಹRT RAನಕ ಹa|ಹ 6 22 12 4 21 12 3 20 16 13 10 7 4 6 4 2 0 24 22 20 18 # 80 ಕ ಟ - 1 ಹ ಜವ 8 - 0 do nನ ಹ - ಹ 8 20 13 0 13 16 4 6 21 10 25 14 3 12 23 6 0 13 0 TR 2 22 1 4 23 ಹPn 1 24 ಕd88೦೦ಕ ಹR ಎ ಉಪ ಹಟ ನನ 18 - Po/p 21 22 22 23 23 24 25 2 24 8 24 2 22 16 10 o ನ ಗ ಹ ೪ ೪ - - - ಹಪ ಹ ಹR & 70 + T 88-ಟ o ನ 88 ನರ ಸ ಹ ೦೭ ೦ನ ಹR ನ ಹ ಹ ಹRR 17 24 19 12 20 9 4 25 24 20 18 12 14 18 20 22 24 10 2 20 8 22 12 22 10 8 17 20 23 0 13 16 25 8 ಉ ರ ನ ನ ಹ ಹR೦ನದ ರಾ0 ರ ನ ಹ ಹ ಹRRog 24 20 16 2 10 18 22 12 6 24 14 20 10 24 7 22 4 20 1 18 2 21 14 V @ A I n -J 0 0 ರ ನ ದ ಹ ದ ಕ ವ ಹ ಹ8RR&R…! 2 14 12 8 6 > 4 21 19 18 16 15 11 10 6 2 4. [12] Use Gaussian elimination to determine the determinant of the matrix: 1 1 4 4 2 -2 8 -8- M = 1 1 1 -1 1 1 -1 -1 You may use software to do the reduction. is linearly independent. 5. [12] Consider two distinct numbers a and b and define the function - 1 1 1 b t b² t² f(t) = det a Answer the following: a) Show that f(t) is a quadratic function. What is the coefficient of t²? b) For what values of t is the matrix invertible? 6. [12] Use the determinant to find the values of the scalar λ for which the matrix A A3 fails to be invertible where, ГО A = 4 L3 2 6 2 305 7. [10] Find the value(s) of k for which the following set of vectors v₁ = (1, 0, 0) v₂ = (k, 1,k) V3 = (1, -2k, 3k + 1)