Matrices: arithmetic and determinants Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Matrices: arithmetic and determinants questions. See exactly how to solve problems on matrices, addition, 2x2, subtraction.

matricesaddition2x2subtractionscalar-multiplicationmultiplication
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Given A=(1234)\mathbf{A}=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} and B=(5−102)\mathbf{B}=\begin{pmatrix}5 & -1 \\ 0 & 2\end{pmatrix}, find A+B\mathbf{A}+\mathbf{B}.

Worked solution

  1. Add corresponding entries

    (1234)+(5−102)\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}+\begin{pmatrix}5 & -1 \\ 0 & 2\end{pmatrix}

    Matrix addition works entry by entry.

  2. Simplify each entry

    (6136)\begin{pmatrix}6 & 1 \\ 3 & 6\end{pmatrix}

    Add the two numbers in each position.

  3. State the resulting matrix

    A+B=(6136)\mathbf{A}+\mathbf{B}=\begin{pmatrix}6 & 1 \\ 3 & 6\end{pmatrix}

    This is A+B\mathbf{A}+\mathbf{B}.

Answer
(6136)\begin{pmatrix}6 & 1 \\ 3 & 6\end{pmatrix}
Question 2
2 markseasy
Given A=(3−241)\mathbf{A}=\begin{pmatrix}3 & -2 \\ 4 & 1\end{pmatrix} and B=(−152−3)\mathbf{B}=\begin{pmatrix}-1 & 5 \\ 2 & -3\end{pmatrix}, find A+B\mathbf{A}+\mathbf{B}.

Worked solution

  1. Add corresponding entries

    (3−241)+(−152−3)\begin{pmatrix}3 & -2 \\ 4 & 1\end{pmatrix}+\begin{pmatrix}-1 & 5 \\ 2 & -3\end{pmatrix}

    Matrix addition works entry by entry.

  2. Simplify each entry

    (236−2)\begin{pmatrix}2 & 3 \\ 6 & -2\end{pmatrix}

    Add the two numbers in each position.

  3. State the resulting matrix

    A+B=(236−2)\mathbf{A}+\mathbf{B}=\begin{pmatrix}2 & 3 \\ 6 & -2\end{pmatrix}

    This is A+B\mathbf{A}+\mathbf{B}.

Answer
(236−2)\begin{pmatrix}2 & 3 \\ 6 & -2\end{pmatrix}
Question 3
2 markseasy
Given A=(7325)\mathbf{A}=\begin{pmatrix}7 & 3 \\ 2 & 5\end{pmatrix} and B=(214−3)\mathbf{B}=\begin{pmatrix}2 & 1 \\ 4 & -3\end{pmatrix}, find A−B\mathbf{A}-\mathbf{B}.

Worked solution

  1. Subtract corresponding entries

    (7325)−(214−3)\begin{pmatrix}7 & 3 \\ 2 & 5\end{pmatrix}-\begin{pmatrix}2 & 1 \\ 4 & -3\end{pmatrix}

    Matrix subtraction works entry by entry.

  2. Simplify each entry

    (52−28)\begin{pmatrix}5 & 2 \\ -2 & 8\end{pmatrix}

    Subtract the second entry from the first in each position.

  3. State the resulting matrix

    A−B=(52−28)\mathbf{A}-\mathbf{B}=\begin{pmatrix}5 & 2 \\ -2 & 8\end{pmatrix}

    This is A−B\mathbf{A}-\mathbf{B}.

Answer
(52−28)\begin{pmatrix}5 & 2 \\ -2 & 8\end{pmatrix}
Question 4
2 markseasy
Given A=(06−24)\mathbf{A}=\begin{pmatrix}0 & 6 \\ -2 & 4\end{pmatrix} and B=(3211)\mathbf{B}=\begin{pmatrix}3 & 2 \\ 1 & 1\end{pmatrix}, find A−B\mathbf{A}-\mathbf{B}.

Worked solution

  1. Subtract corresponding entries

    (06−24)−(3211)\begin{pmatrix}0 & 6 \\ -2 & 4\end{pmatrix}-\begin{pmatrix}3 & 2 \\ 1 & 1\end{pmatrix}

    Matrix subtraction works entry by entry.

  2. Simplify each entry

    (−34−33)\begin{pmatrix}-3 & 4 \\ -3 & 3\end{pmatrix}

    Subtract the second entry from the first in each position.

  3. State the resulting matrix

    A−B=(−34−33)\mathbf{A}-\mathbf{B}=\begin{pmatrix}-3 & 4 \\ -3 & 3\end{pmatrix}

    This is A−B\mathbf{A}-\mathbf{B}.

Answer
(−34−33)\begin{pmatrix}-3 & 4 \\ -3 & 3\end{pmatrix}
Question 5
2 markseasy
Given A=(2−135)\mathbf{A}=\begin{pmatrix}2 & -1 \\ 3 & 5\end{pmatrix}, find 3A3\mathbf{A}.

Worked solution

  1. Multiply every entry by the scalar

    3(2−135)3\begin{pmatrix}2 & -1 \\ 3 & 5\end{pmatrix}

    Scalar multiplication scales each entry.

  2. Evaluate the scaled entries

    (6−3915)\begin{pmatrix}6 & -3 \\ 9 & 15\end{pmatrix}

    Each entry is multiplied by 33.

  3. State the scaled matrix

    3A=(6−3915)3\mathbf{A}=\begin{pmatrix}6 & -3 \\ 9 & 15\end{pmatrix}

    This is 3A3\mathbf{A}.

Answer
(6−3915)\begin{pmatrix}6 & -3 \\ 9 & 15\end{pmatrix}

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