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Part III: The Power Stuff: Advanced Techniques
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Add 1 times the first row to the third row and place the result in the third row, like so: J 1 0 0 N J 1N OK O K K 0 1 1O K 5 O OK O K K 0 1 -1O K -1O PL P L Now add 1 times the second row to the third, placing the result in the third row: J1 0 0 N J 1N OK O K K0 1 1O K 5 O OK O K K 0 0 - 2O K - 6 O PL P L Finally, you divide the third row by 2: J 1 0 0 N J 1N OK O K K 0 1 1O K 5 O K OK O K 0 0 1O K 3 O PL P L Note that you now have a triangular matrix on the bottom left; in other words, the bottom left triangle s elements all equal zero This triangle simplifies matters considerably, because this new augmented matrix now represents this system: x=1 y+z=5 z=3 And from the latter two equations, finding the solution for y is pretty trivial: y = 2 Now you ve solved for all three variables Nice work
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Examining matrices
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Did you know that you can name matrices Yup, that s right In matrix terms, a name is denoted in bold For instance, you can name the following matrix, which contains both real and complex elements, A (imaginative, I know): J 1 2 + 2i N O A=K K 3 + 3i 4 O L P
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12: Tackling Systems of First Order Linear Differential Equations
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The transpose of a matrix swaps its rows and columns; in other words, the rows become the columns, and the columns become the rows Here s what the transpose of A (called AT ) looks like: J 1 3 + 3i N O AT = K K 2 + 2i 4 O L P You can also define the complex conjugate of a matrix To denote a complex conjugate, you simply add a line above the name of a matrix The complex conjugate of a matrix is simply the complex conjugate element by element of the matrix Here s the complex conjugate of A, where you flip the sign of the imaginary part: J 1 2 - 2i N O A=K K 3 - 3i 4 O L P The adjoint of a matrix is the transpose of the complex conjugate of the matrix You denote an adjoint by adding an asterisk (*) to the name of the matrix Here s what A* looks like: J 1 3 - 3i N O A* = K K 2 - 2i 4 O L P
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Mastering Matrix Operations
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After you re familiar with the basics of matrices, you can take the next step and start working with a variety of matrix operations, such as addition, subtraction, multiplication, and other fun stuff I explain what you need to know in the following sections
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Equality
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Two matrices are considered equal if every element in the first matrix is equal to the corresponding element in the other matrix For instance, if: J1 2 N O A=K K 3 4O L P And: J1 2 N O B=K K 3 4O L P then A = B Easy enough, right
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Part III: The Power Stuff: Advanced Techniques
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If you need to, you can add two matrices together Doing so involves adding the elements at corresponding positions in the two matrices For example, if: J1 2 N J5 6 N O K O A=K K 3 4 O and B = K 7 8 O L P L P then: J1 2 N J5 6 N J 6 8 N O K O K O A+B=K K 3 4 O + K 7 8 O = K 10 12O L P L P L P And just so you know: A + B = B + A
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