1001 Algebra Problems.PDF

840. a.Two matrices are equal if and only if each pair of corresponding entries is the same. Note that
     . Therefore, the equation is equivalent to ,

which occurs only when x= 3.

841. d.Two matrices are equal if and only if each pair of corresponding entries are the same. Therefore, both
     x^2 = 4 and 3x= 6 must hold simultaneously. The solutions of the first equation are x= –2 and 2, but
     onlyx= 2 also satisfies the second equation. Thus, we conclude that the only x-value that makes the
     equality true is x= 2.


842. d.Two matrices are equal if and only if each pair of corresponding entries are the same. Observe that

and. The two matrices are equal if and only if 3x= – 6y.

This implies that for any given real value ofx, if we choose y= –^12 x, the ordered pair (x,y) makes the

equality true. Since there are infinitely many possible values ofx, we conclude that infinitely many

ordered pairs make this equality true.

843. b.First, simplify the left side of the equation:

Equating corresponding entries reveals that the following equations must hold simultaneously: –4x+ 12

= 2xand –2y– 8 = y. Solving these equations, we find that x= 2 and y= –^83 . Hence, the ordered pair that

makes the equality true is (2, –^83 ).

844. c.Simplifying the left side of the equation yields the equivalent equation. Equating

corresponding entries reveals that the following equations must hold simultaneously: –x= 3x,–y= 4y,

and –z= –2z. Solving these equations, we find that x = y = z =0. Hence, the ordered triple that makes

the equality true is (0,0,0).

845. a.The sum of two matrices is defined only when both matrices have the same number of rows and
     columns (i.e., the same dimensions); the resulting matrix is one with the same number of rows and
     columns. Statement ais true.

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ANSWERS & EXPLANATIONS–