Matrices and Matrix Operations: Precalculus Math Worksheet Answer Key – PDF Download

Precalculus Matrices Worksheet Answer Key: problems 1–5

Problem 1: Dimensions and entries

\(G\) has 3 rows and 4 columns, so its dimensions are \(3\times 4\).

1. Row 2, column 3: \(-3\)
2. Row 3, column 4: \(-2\)
3. The entry 7 is in row 1, column 3.

Problem 2: Organize data in a matrix

The matrix has 3 rows and 2 columns, so it is \(3\times 2\).

Muffins are row 3: \(30+21=\) 51 muffins.

Problem 3: Add matrices

Add matching entries: \(3+(-1)=2\), \(-2+4=2\), \(5+2=7\), \(0+6=6\).

Problem 4: Subtract matrices

Subtract matching entries: row 1 is \(7-4=3\), \(2-(-1)=3\), \(-3-2=-5\); row 2 is \(0-3=-3\), \(5-5=0\), \(1-(-6)=7\).

Problem 5: Scalar multiples

Multiply every entry by the scalar.

Precalculus Matrices Worksheet Answer Key: problems 6–10

Problem 6: Combine operations

First \(2A\) has entries 6, \(-4\), 10, 0. Then \(3B\) has entries \(-3\), 12, 6, 18. Subtract entry by entry: \(6-(-3)=9\), \(-4-12=-16\), \(10-6=4\), \(0-18=-18\).

Problem 7: Solve for unknowns

Equal matrices have equal entries. Top left: \(x+3=7\), so \(x=4\). Top right: \(2+y=5\), so \(y=3\).

Check with the bottom row: \(5+x=9\) and \(y+1=4\). Both hold, so \(x=4\) and \(y=3\).

Problem 8: Is the product defined?

1. Columns of \(A\) = 3 = rows of \(B\): defined, \(2\times 4\)
2. Columns of \(A\) = 2 but rows of \(B\) = 3: not defined
3. Inner numbers match (3 and 3): defined, \(1\times 1\)
4. Inner numbers match (1 and 1): defined, \(3\times 3\)

Problem 9: Multiply 2 × 2 matrices

• \(c_{11}=1\cdot 4+3\cdot (-2)=-2\)
• \(c_{12}=1\cdot 0+3\cdot 5=15\)
• \(c_{21}=2\cdot 4+(-1)\cdot (-2)=10\)
• \(c_{22}=2\cdot 0+(-1)\cdot 5=-5\)

Problem 10: Multiply a 2 × 3 by a 3 × 2

\(R\) is \(2\times 3\) and \(S\) is \(3\times 2\), so \(RS\) is \(2\times 2\).

• \(c_{11}=1\cdot 2+0\cdot 0+2\cdot 4=10\)
• \(c_{12}=1\cdot 1+0\cdot (-2)+2\cdot 3=7\)
• \(c_{21}=(-1)\cdot 2+3\cdot 0+1\cdot 4=2\)
• \(c_{22}=(-1)\cdot 1+3\cdot (-2)+1\cdot 3=-4\)

Precalculus Matrices Worksheet Answer Key: problems 11–15

Problem 11: Matrix times a column

The result is \(2\times 1\). Row 1: \(2\cdot 5+(-1)\cdot 2=8\). Row 2: \(3\cdot 5+4\cdot 2=23\).

Problem 12: Compare AB and BA

\(AB\): \(c_{11}=1\cdot 3+2\cdot 1=5\), \(c_{12}=1\cdot 0+2\cdot 1=2\), \(c_{21}=0\cdot 3+1\cdot 1=1\), \(c_{22}=0\cdot 0+1\cdot 1=1\).

\(BA\): \(d_{11}=3\cdot 1+0\cdot 0=3\), \(d_{12}=3\cdot 2+0\cdot 1=6\), \(d_{21}=1\cdot 1+1\cdot 0=1\), \(d_{22}=1\cdot 2+1\cdot 1=3\).

The matrices differ, so \(AB\neq BA\).

Problem 13: Identity and zero matrices

1. Multiplying by the identity changes nothing: \(AI=A\), the matrix with rows 4, \(-3\) and 2, 7.
2. Adding the zero matrix changes nothing: \(A+O=A\).
3. Every row-by-column sum is 0: \(AO=O\), the \(2\times 2\) zero matrix.
4. \(-A\) has entries \(-4,3,-2,-7\), so the sum is \(O\).

Problem 14: Check the distributive property

Left side: \(B+C\) has rows 3, 0 and \(-1\), 3. Then \(A(B+C)\):

• \(c_{11}=1\cdot 3+2\cdot (-1)=1\)
• \(c_{12}=1\cdot 0+2\cdot 3=6\)
• \(c_{21}=3\cdot 3+0\cdot (-1)=9\)
• \(c_{22}=3\cdot 0+0\cdot 3=0\)

Right side: \(AB\) has rows 2, 1 and 6, \(-3\); \(AC\) has rows \(-1\), 5 and 3, 3. Adding gives rows 1, 6 and 9, 0.

The two sides are equal, so \(A(B+C)=AB+AC\).

Problem 15: Multiply 3 × 3 matrices

Each entry is a row of \(X\) times a column of \(Y\). Row 1 of the product: \(1\cdot 2+0\cdot 1+2\cdot 0=2\), \(1\cdot 1+0\cdot 0+2\cdot 2=5\), \(1\cdot 0+0\cdot 1+2\cdot 1=2\). Row 2: \(1\), \(-2\), \(0\). Row 3: \(7\), \(3\), \(1\).

Precalculus Matrices Worksheet Answer Key: problems 16–20

Problem 16: Find determinants

1. \(3\cdot 4-5\cdot 2=12-10=\) 2
2. \((-2)(-3)-6\cdot 1=6-6=\) 0
3. \(0\cdot 7-4\cdot(-3)=0+12=\) 12
4. \(5\cdot 3-(-1)\cdot 2=15+2=\) 17

Problem 17: Find k for a singular matrix

1. \(\det=3k-12\). Set \(3k-12=0\): \(k=4\).
2. \(\det=k^{2}-16\). Set \(k^{2}-16=0\): \(k=4\) or \(k=-4\).

Problem 18: Find a 2 × 2 inverse

1. \(\det=3\cdot 2-1\cdot 5=1\). Swap 3 and 2, negate 1 and 5, and multiply by \(\frac{1}{1}\): the inverse has rows 2, \(-1\) and \(-5\), 3.
2. \(\det=4\cdot 2-2\cdot 3=2\). The swapped matrix has rows 2, \(-2\) and \(-3\), 4. Multiply by \(\frac{1}{2}\): rows 1, \(-1\) and \(-\frac{3}{2}\), 2.

Problem 19: Verify an inverse

\(AB\): \(c_{11}=3\cdot 2+1\cdot (-5)=1\), \(c_{12}=3\cdot (-1)+1\cdot 3=0\), \(c_{21}=5\cdot 2+2\cdot (-5)=0\), \(c_{22}=5\cdot (-1)+2\cdot 3=1\).

\(BA\): \(d_{11}=2\cdot 3+(-1)\cdot 5=1\), \(d_{12}=2\cdot 1+(-1)\cdot 2=0\), \(d_{21}=(-5)\cdot 3+3\cdot 5=0\), \(d_{22}=(-5)\cdot 1+3\cdot 2=1\).

Since \(AB=BA=I\), \(B=A^{-1}\).

Problem 20: School store revenue

\(S\) is \(2\times 3\) and \(P\) is \(3\times 1\), so \(SP\) is \(2\times 1\).

• Monday: \(12(0.50)+8(0.75)+5(2.25)=6+6+11.25=23.25\)
• Tuesday: \(15(0.50)+6(0.75)+9(2.25)=7.5+4.5+20.25=32.25\)

Revenue was \$23.25 on Monday and \$32.25 on Tuesday.

Precalculus Matrices Worksheet Answer Key: problems 21–23

Problem 21: Farmers market inventory

Add matching entries, then multiply by 2. For example, Saturday apples: \(40+10=50\), and doubled \(2\cdot 50=100\).

Problem 22: Encode and decode a message

Coded: row 1 is \(2\cdot 7+1\cdot 12=26\) and row 2 is \(3\cdot 7+2\cdot 12=45\).

\(\det K=2\cdot 2-1\cdot 3=1\), so \(K^{-1}\) has rows 2, \(-1\) and \(-3\), 2. Decode: \(2\cdot 26+(-1)\cdot 45=7\) and \((-3)\cdot 26+2\cdot 45=12\). The original message is \((7,12)\).

Problem 23: Does the matrix have an inverse?

\(\det N=6\cdot 6-9\cdot 4=36-36=0\). A 2 × 2 matrix has an inverse only when its determinant is not zero, so Noah is wrong: \(N\) has no inverse. Positive entries do not guarantee a nonzero determinant.

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