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Agent Q:Glad you asked, Pond.We're going to use the matrix on the screen and the master key to find the code.Let's call the digits we need a, b, c.To find these digits, we can form the following equation:A(1 0) + B(0 2) + C(0 0) = (4 2) (0 0) (2 0) (0 1) (2 6)
Agent Q:We can simplify the left side:A(1 0) + B(0 2) + C(0 0) = (4 2) (0 0) (2 0) (0 1) (2 6)First, we'll multiply the letters into the matrices(A 0) + (0 2B) + (0 0) = (4 2)(0 0) (2B 0) (0 C) (2 6)Then, we can add up the terms in each cell of the matrices(A 2B) = (4 2)(2B C) (2 6)Clearly, A is equal to 4, B is equal to 1, and C is equal to 6.
Agent Pond:Using the 2x2 matrix on the lock and the master key that agent Q gave us, we need to find the 3 digits passcode to open the door.2x2 matrix on the lock: (2 6) (6 1)Master key: (1 0), (0 2), (0 0) (0 1) (2 0), (0 1)Hmmmm... I don't fully remember how to solve this...'4 1 6'' 2 3 1'
These vectors amke up a basis that can be used to make any 2x2 matrix that are part of a specific vector space.
Agent Pond:So first we have to write out the equation where we multiply the basis with the letters and set it equal to the matrix.A(__ 0) + B(0 2) + C(0 __) = (__ __) (__ 0) (__ __) (0 __) (__ __)Then we have to multiply the letters in.(__ 0) + (0 2__) + (0 __) = (__ __)(__ 0) (____ 0) (0 __) (__ __)And now we have to add the terms together...(__ ____) = (__ __)(____ __) (__ __)... and now we can solve it!A = __________, B = __________, C = __________

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