1. No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a [latex]2\\times 2[\/latex] matrix and the second is a [latex]2\\times 3[\/latex] matrix. [latex]\\left[\\begin{array}{cc}1& 2\\\\ 3& 4\\end{array}\\right]+\\left[\\begin{array}{ccc}6& 5& 4\\\\ 3& 2& 1\\end{array}\\right][\/latex] has no sum.<\/p>\n
2. No. Matrices can only be multiplied if the inner dimensions are equal. For example, a 2×3 matrix cannot be multiplied by another 2×3 matrix since 3 does not equal 2.<\/p>\n
3. [latex]\\left[\\begin{array}{cc}11& 19\\\\ 15& 94\\\\ 17& 67\\end{array}\\right][\/latex]<\/p>\n
4. [latex]\\left[\\begin{array}{cc}-4& 2\\\\ 8& 1\\end{array}\\right][\/latex]<\/p>\n
5. Undidentified; dimensions do not match<\/p>\n
6. [latex]\\left[\\begin{array}{cc}9& 27\\\\ 63& 36\\\\ 0& 192\\end{array}\\right][\/latex]<\/p>\n
7. [latex]\\left[\\begin{array}{cccc}-64& -12& -28& -72\\\\ -360& -20& -12& -116\\end{array}\\right][\/latex]<\/p>\n
8. [latex]\\left[\\begin{array}{ccc}1,800& 1,200& 1,300\\\\ 800& 1,400& 600\\\\ 700& 400& 2,100\\end{array}\\right][\/latex]<\/p>\n
9. [latex]\\left[\\begin{array}{cc}20& 102\\\\ 28& 28\\end{array}\\right][\/latex]<\/p>\n
10. [latex]\\left[\\begin{array}{ccc}60& 41& 2\\\\ -16& 120& -216\\end{array}\\right][\/latex]<\/p>\n
11. [latex]\\left[\\begin{array}{ccc}-68& 24& 136\\\\ -54& -12& 64\\\\ -57& 30& 128\\end{array}\\right][\/latex]<\/p>\n
12. Undefined; dimensions do not match.<\/p>\n
13. [latex]\\left[\\begin{array}{ccc}-8& 41& -3\\\\ 40& -15& -14\\\\ 4& 27& 42\\end{array}\\right][\/latex]<\/p>\n
14. [latex]\\left[\\begin{array}{ccc}-840& 650& -530\\\\ 330& 360& 250\\\\ -10& 900& 110\\end{array}\\right][\/latex]<\/p>\n
15. [latex]\\left[\\begin{array}{cc}-350& 1,050\\\\ 350& 350\\end{array}\\right][\/latex]<\/p>\n
16. Undefined; inner dimensions do not match.<\/p>\n
17. [latex]\\left[\\begin{array}{cc}1,400& 700\\\\ -1,400& 700\\end{array}\\right][\/latex]<\/p>\n
18. [latex]\\left[\\begin{array}{cc}332,500& 927,500\\\\ -227,500& 87,500\\end{array}\\right][\/latex]<\/p>\n
19. [latex]\\left[\\begin{array}{cc}490,000& 0\\\\ 0& 490,000\\end{array}\\right][\/latex]<\/p>\n
20. [latex]\\left[\\begin{array}{ccc}-2& 3& 4\\\\ -7& 9& -7\\end{array}\\right][\/latex]<\/p>\n
21. [latex]\\left[\\begin{array}{ccc}-4& 29& 21\\\\ -27& -3& 1\\end{array}\\right][\/latex]<\/p>\n
22. [latex]\\left[\\begin{array}{ccc}-3& -2& -2\\\\ -28& 59& 46\\\\ -4& 16& 7\\end{array}\\right][\/latex]<\/p>\n
23. [latex]\\left[\\begin{array}{ccc}1& -18& -9\\\\ -198& 505& 369\\\\ -72& 126& 91\\end{array}\\right][\/latex]<\/p>\n
24. [latex]\\left[\\begin{array}{cc}0& 1.6\\\\ 9& -1\\end{array}\\right][\/latex]<\/p>\n
25. [latex]\\left[\\begin{array}{ccc}2& 24& -4.5\\\\ 12& 32& -9\\\\ -8& 64& 61\\end{array}\\right][\/latex]<\/p>\n
1. Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.<\/p>\n
2. [latex]\\left[\\left.\\begin{array}{rrrr}\\hfill 0& \\hfill & \\hfill 16& \\hfill \\\\ \\hfill 9& \\hfill & \\hfill -1& \\hfill \\end{array}\\right\\rvert\\begin{array}{rr}\\hfill & \\hfill 4\\\\ \\hfill & \\hfill 2\\end{array}\\right][\/latex]<\/p>\n
3. [latex]\\left[\\left.\\begin{array}{rrrrrr}\\hfill 1& \\hfill & \\hfill 5& \\hfill & \\hfill 8& \\hfill \\\\ \\hfill 12& \\hfill & \\hfill 3& \\hfill & \\hfill 0& \\hfill \\\\ \\hfill 3& \\hfill & \\hfill 4& \\hfill & \\hfill 9& \\hfill \\end{array}\\right\\rvert\\begin{array}{rr}\\hfill & \\hfill 16\\\\ \\hfill & \\hfill 4\\\\ \\hfill & \\hfill -7\\end{array}\\right][\/latex]<\/p>\n
4. [latex]\\begin{array}{l}-2x+5y=5\\\\ 6x - 18y=26\\end{array}[\/latex]<\/p>\n
5. [latex]\\begin{array}{l}3x+2y=13\\\\ -x - 9y+4z=53\\\\ 8x+5y+7z=80\\end{array}[\/latex]<\/p>\n
6. [latex]\\begin{array}{l}4x+5y - 2z=12\\hfill \\\\ \\text{ }y+58z=2\\hfill \\\\ 8x+7y - 3z=-5\\hfill \\end{array}[\/latex]<\/p>\n
7. No solutions<\/p>\n
8. [latex]\\left(-1,-2\\right)[\/latex]<\/p>\n
9. [latex]\\left(6,7\\right)[\/latex]<\/p>\n
10. [latex]\\left(\\frac{1}{5},\\frac{1}{2}\\right)[\/latex]<\/p>\n
11. [latex]\\left(x,\\frac{4}{15}\\left(5x+1\\right)\\right)[\/latex]<\/p>\n
12. [latex]\\left(\\frac{196}{39},-\\frac{5}{13}\\right)[\/latex]<\/p>\n
13. [latex]\\left(31,-42,87\\right)[\/latex]<\/p>\n
14. [latex]\\left(\\frac{21}{40},\\frac{1}{20},\\frac{9}{8}\\right)[\/latex]<\/p>\n
15. [latex]\\left(\\frac{18}{13},\\frac{15}{13},-\\frac{15}{13}\\right)[\/latex]<\/p>\n
16. [latex]\\left(x,y,\\frac{1}{2}\\left(1 - 2x - 3y\\right)\\right)[\/latex]<\/p>\n
17. [latex]\\left(125,-25,0\\right)[\/latex]<\/p>\n
18. [latex]\\left(8,1,-2\\right)[\/latex]<\/p>\n
19. 860 red velvet, 1,340 chocolate<\/p>\n
20. 4% for account 1, 6% for account 2<\/p>\n
21. $126<\/p>\n
22. Banana was 3%, pumpkin was 7%, and rocky road was 2%<\/p>\n
23. 100 almonds, 200 cashews, 600 pistachios<\/p>\n
1. No, because [latex]ad[\/latex] and [latex]bc[\/latex] are both 0, so [latex]ad-bc=0[\/latex], which requires us to divide by 0 in the formula.<\/p>\n
2. [latex]AB=BA=\\left[\\begin{array}{cc}1& 0\\\\ 0& 1\\end{array}\\right]=I[\/latex]<\/p>\n
3. [latex]AB=BA=\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right]=I[\/latex]<\/p>\n
4. [latex]\\frac{1}{29}\\left[\\begin{array}{cc}9& 2\\\\ -1& 3\\end{array}\\right][\/latex]<\/p>\n
5. There is no inverse<\/p>\n
6. [latex]\\frac{4}{7}\\left[\\begin{array}{cc}0.5& 1.5\\\\ 1& -0.5\\end{array}\\right][\/latex]<\/p>\n
7. [latex]\\frac{1}{17}\\left[\\begin{array}{ccc}-5& 5& -3\\\\ 20& -3& 12\\\\ 1& -1& 4\\end{array}\\right][\/latex]<\/p>\n
8. [latex]\\left[\\begin{array}{ccc}18& 60& -168\\\\ -56& -140& 448\\\\ 40& 80& -280\\end{array}\\right][\/latex]<\/p>\n
9. [latex]\\left(-5,6\\right)[\/latex]<\/p>\n
10. [latex]\\left(\\frac{1}{3},-\\frac{5}{2}\\right)[\/latex]<\/p>\n
11. [latex]\\left(5,0,-1\\right)[\/latex]<\/p>\n
12. [latex]\\frac{1}{690}\\left(65,-1136,-229\\right)[\/latex]<\/p>\n
13. [latex]\\left(-\\frac{37}{30},\\frac{8}{15}\\right)[\/latex]<\/p>\n
14. [latex]\\left(\\frac{10}{123},-1,\\frac{2}{5}\\right)[\/latex]<\/p>\n
15. [latex]\\frac{1}{2}\\left[\\begin{array}{rrrr}\\hfill 2& \\hfill 1& \\hfill -1& \\hfill -1\\\\ \\hfill 0& \\hfill 1& \\hfill 1& \\hfill -1\\\\ \\hfill 0& \\hfill -1& \\hfill 1& \\hfill 1\\\\ \\hfill 0& \\hfill 1& \\hfill -1& \\hfill 1\\end{array}\\right][\/latex]<\/p>\n
16. [latex]\\frac{1}{39}\\left[\\begin{array}{rrrr}\\hfill 3& \\hfill 2& \\hfill 1& \\hfill -7\\\\ \\hfill 18& \\hfill -53& \\hfill 32& \\hfill 10\\\\ \\hfill 24& \\hfill -36& \\hfill 21& \\hfill 9\\\\ \\hfill -9& \\hfill 46& \\hfill -16& \\hfill -5\\end{array}\\right][\/latex]<\/p>\n
17. 50% oranges, 25% bananas, 20% apples<\/p>\n
18. 10 straw hats, 50 beanies, 40 cowboy hats<\/p>\n
19. Tom ate 6, Joe ate 3, and Albert ate 3.<\/p>\n
20. 124 oranges, 10 lemons, 8 pomegranates<\/p>\n
1. A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product\u2014even if it does end up being 0.<\/p>\n
2. The inverse does not exist.<\/p>\n
3. [latex]-2[\/latex]<\/p>\n
4. [latex]7[\/latex]<\/p>\n
5. [latex]0[\/latex]<\/p>\n
6. [latex]3[\/latex]<\/p>\n
7. [latex]224[\/latex]<\/p>\n
8. [latex]-17.03[\/latex]<\/p>\n
9. [latex]\\left(1,1\\right)[\/latex]<\/p>\n
10. [latex]\\left(\\frac{1}{2},\\frac{1}{3}\\right)[\/latex]<\/p>\n
11. [latex]\\left(15,12\\right)[\/latex]<\/p>\n
12. [latex]\\left(1,3,2\\right)[\/latex]<\/p>\n
13. [latex]\\left(-1,0,3\\right)[\/latex]<\/p>\n
14. [latex]\\left(\\frac{1}{2},1,2\\right)[\/latex]<\/p>\n
15. Infinite solutions<\/p>\n
16. $7,000 in first account, $3,000 in second account.<\/p>\n
17. 120 children, 1,080 adult<\/p>\n
18. 4 gal yellow, 6 gal blue<\/p>\n
19. 13 green tomatoes, 17 red tomatoes<\/p>\n
20. Strawberries 18%, oranges 9%, kiwi 10%<\/p>\n","protected":false},"author":13,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/328"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/328\/revisions"}],"predecessor-version":[{"id":404,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/328\/revisions\/404"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/328\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=328"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=328"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=328"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=328"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}