Algebra.Com's Answer #842308 by ikleyn(52835)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "I don't know how to format matrices on here, so I will write the top row (horizontal) first and then the bottom row.
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document.write( "Consider the following matrices:
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document.write( "A = [4, -1][-6, -3]
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document.write( "B = XA, where X is the 2x2 matrix that dilates vectors by a factor of 2
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document.write( "C = The 3x3 matrix that reflects vectors across the plane x + y + z = 1
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document.write( "D = [4, -1, 5][0, -3, -3][0, 0, -5]
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document.write( "Calculate the four absolute values |det(A)|, |det(B)|, |det(C)|, and |det(D)|.
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document.write( "(a) Matrix A is \r\n" );
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document.write( " A =  .\r\n" );
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document.write( " Its determinant is det(A) = 4*(-3) - (-6)*(-1) = -12 - 6 = -18.\r\n" );
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document.write( " The absolute value of the determinant A is |det(A)| = |-18| = 18.\r\n" );
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document.write( " Usually, at normal pedagogic process, a student learns the notion/(the conception) \r\n" );
document.write( " of a matrix and its determinant and how to calculate it in the same day \r\n" );
document.write( " (from the same lecture of a professor or from a textbook).\r\n" );
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document.write( "(b) Matrix X, which is 2x2 matrix that dilates vecors by a factor of 2 is the diagonal matrix\r\n" );
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document.write( " X =  with the scalar elements of 2 in its diagonal.\r\n" );
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document.write( " Multiplication by matrix X from the left (XA) acts on matrix A by multiplying all elements of A \r\n" );
document.write( " by the scalar of 2. So, the product XA is\r\n" );
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document.write( " XA =  .\r\n" );
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document.write( " You can calculate XA directly by making direct multiplication, or you can restore the product XA\r\n" );
document.write( " using the rule above.\r\n" );
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document.write( " Next, if the rows of the matrix are multiplied by 2, then, according to the rules of determinant,\r\n" );
document.write( " det(XA) is equal to det(A) multiplied by 2*2 = 4.\r\n" );
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document.write( " Using this property of determinants, you can get det(XA) = 4*det(A) = 4*(-18) = -72 and \r\n" );
document.write( " |det(XA)| = |-72| = 72.\r\n" );
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document.write( " Alternatively, you can calculate det(XA) directly, based on the formal rule of calculating determinants.\r\n" );
document.write( " Surely, you will get the same value.\r\n" );
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document.write( "(c) Determinant of this reflection matrix can be found without calculations,\r\n" );
document.write( " and even without writing of this matrix in explicit form.\r\n" );
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document.write( " Determinant of this matrix is +/-1, since this geometric operation of reflection conserves the volume.\r\n" );
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document.write( " Therefore, the absolute value of the determinant of this matrix is |det(C)| = 1.\r\n" );
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document.write( "(d) Matrix D is an upper triangular matrix.\r\n" );
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document.write( " The determinant of such matrix is the product of its diagonal elements det(D) = 4*(-3)*(-5) = 4*15 = 60.\r\n" );
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document.write( " Therefore, |det(D)| = |60| = 60.\r\n" );
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document.write( "Solved. I answered all your questions.\r
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document.write( "Level of question (a) is 0 (zero): a student should know it from the professor lecture or from a textbook
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document.write( "in the first day as the matrices and determinants are explained and introduced.\r
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document.write( "Level of question (b) is between 0.5 and 1, which means that from one half of
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document.write( "the thought to one whole thought is needed \r
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document.write( "Level of question (c) is between 2 and 2.5, which means that you may learn it either from an
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document.write( "advanced professor lecture or from an advanced textbook.\r
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document.write( "Level of question (d) is 1, which means that almost everybody should know the answer and the method,
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document.write( "but usually it is learned not at the first day, but somewhen later.\r
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