Matrices

21 Grade 9 Mathematics questions on Matrices, with worked marking schemes. 21 questions are available on this topic in full.

1. Question 1

   1 mark

   Find the determinant of $\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$.

   A. 10

   B. −10

   C. 14

   D. 8

   (1 mark)

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2. Question 2

   1 mark

   The determinant of $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ = ___

   A. ac − bd

   B. ad − bc

   C. ab − cd

   D. ad + bc

   (1 mark)

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3. Question 3

   1 mark

   The 2×2 identity matrix I₂ = ___

   A. $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

   B. $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

   C. $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

   D. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

   (1 mark)

   Marking scheme

   D. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ — the identity has $1$s down the leading diagonal and $0$s elsewhere, so that multiplying by it leaves any matrix unchanged. C has the $1$s on the other diagonal.

4. Question 4

   4 marks

   If A = $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and B = $\left[\begin{array}{cc}2& 0\\ 1& 3\end{array}\right]$, compute A + B and A − B.

   (4 marks)

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5. Question 5

   1 mark

   A 2×3 matrix has ___ rows and ___ columns.

   A. 3 rows, 2 columns

   B. 1 row, 6 columns

   C. 2 rows, 3 columns

   D. 6 rows, 1 column

   (1 mark)

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6. Question 6

   1 mark

   A matrix with the same number of rows and columns is called a ___ matrix.

   A. Square

   B. Rectangular

   C. Identity

   D. Zero

   (1 mark)

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7. Question 7

   1 mark

   If A = $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and B = $\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$, what is A + B?

   A. $\left[\begin{array}{cc}4& 4\\ 4& 4\end{array}\right]$

   B. $\left[\begin{array}{cc}6& 8\\ 10& 11\end{array}\right]$

   C. $\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$

   D. $\left[\begin{array}{cc}5& 12\\ 21& 32\end{array}\right]$

   (1 mark)

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8. Question 8

   1 mark

   If 2A = $\left[\begin{array}{cc}4& 6\\ 8& 10\end{array}\right]$, what is matrix A?

   A. $\left[\begin{array}{cc}4& 6\\ 8& 10\end{array}\right]$

   B. $\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$

   C. $\left[\begin{array}{cc}8& 12\\ 16& 20\end{array}\right]$

   D. $\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$

   (1 mark)

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9. Question 9

   4 marks

   Multiply: $\left[\begin{array}{cc}2& 1\\ 0& 3\end{array}\right]$ × $\left[\begin{array}{cc}1& 4\\ 2& 0\end{array}\right]$

   (4 marks)

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10. Question 10

    3 marks

    Find det A where A = $\left[\begin{array}{cc}4& 2\\ 6& 3\end{array}\right]$. Is A singular or non-singular?

    (3 marks)

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11. Question 11

    1 mark

    What is the trace (sum of main diagonal elements) of $\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$?

    A. 5

    B. 9

    C. 7

    D. 3

    (1 mark)

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12. Question 12

    3 marks

    Find x and y given: $\left[\begin{array}{cc}x& 3\\ 2& y\end{array}\right]$ + $\left[\begin{array}{cc}1& -1\\ 0& 4\end{array}\right]$ = $\left[\begin{array}{cc}5& 2\\ 2& 7\end{array}\right]$

    (3 marks)

    Marking scheme

    Two matrices are added entry by entry, and two matrices are equal only when every matching entry is equal. (1)

    The top-left entries give $x+1=5$, so $x=4$. (1)

    The bottom-right entries give $y+4=7$, so $y=3$. (1)

    Final answer: $x=4$ and $y=3$. Check the other entries: $3+(-1)=2$ and $2+0=2$.

13. Question 13

    1 mark

    Matrix A = $\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$. What is the element in row 2, column 1?

    A. 2

    B. 4

    C. 3

    D. 5

    (1 mark)

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14. Question 14

    1 mark

    A = $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, B = $\left[\begin{array}{cc}3& 4\\ 5& 6\end{array}\right]$. What is AB?

    A. $\left[\begin{array}{cc}3& 5\\ 4& 6\end{array}\right]$

    B. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

    C. $\left[\begin{array}{cc}4& 4\\ 5& 7\end{array}\right]$

    D. $\left[\begin{array}{cc}3& 4\\ 5& 6\end{array}\right]$

    (1 mark)

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15. Question 15

    1 mark

    Given $A=\left(\begin{array}{cc}1& 2\\ 3& 1\end{array}\right)$ and $B=\left(\begin{array}{cc}2& 0\\ 1& 3\end{array}\right)$, find the product $AB$.

    A. $\left(\begin{array}{cc}2& 4\\ 10& 5\end{array}\right)$

    B. $\left(\begin{array}{cc}2& 0\\ 3& 3\end{array}\right)$

    C. $\left(\begin{array}{cc}4& 6\\ 7& 3\end{array}\right)$

    D. $\left(\begin{array}{cc}3& 2\\ 4& 4\end{array}\right)$

    (1 mark)

    Marking scheme

    C. $\left(\begin{array}{cc}4& 6\\ 7& 3\end{array}\right)$ — each entry is a row of $A$ combined with a column of $B$. The top-left is $(1)(2)+(2)(1)=4$ and the top-right is $(1)(0)+(2)(3)=6$; the second row gives $(3)(2)+(1)(1)=7$ and $(3)(0)+(1)(3)=3$. Multiplying matching positions instead would be wrong.

16. Question 16

    1 mark

    Find det $\left[\begin{array}{cc}5& 2\\ 3& 1\end{array}\right]$.

    A. −1

    B. 11

    C. 1

    D. −11

    (1 mark)

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17. Question 17

    3 marks

    Write the system 2x + y = 7, x + 3y = 11 as a matrix equation AX = B.

    (3 marks)

    Marking scheme

    A = $\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]$, X = $\left[\begin{array}{c}x\\ y\end{array}\right]$, B = $\left[\begin{array}{c}7\\ 11\end{array}\right]$

    Equation: $\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]$ $\left[\begin{array}{c}x\\ y\end{array}\right]$ = $\left[\begin{array}{c}7\\ 11\end{array}\right]$

18. Question 18

    1 mark

    The product of a 2×3 and a 3×2 matrix has dimensions ___

    A. 3×3

    B. 2×2

    C. 3×2

    D. 2×3

    (1 mark)

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19. Question 19

    1 mark

    How many elements does a 3×4 matrix have?

    A. 16

    B. 7

    C. 12

    D. 9

    (1 mark)

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20. Question 20

    4 marks

    Given A = $\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$, find the determinant and write the inverse A⁻¹.

    (4 marks)

    Marking scheme

    For $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ the determinant is $ad-bc$. (1)

    Here $\det A=(2)(3)-(5)(1)=6-5=1$. (1)

    The inverse is $A^{-1}=\frac{1}{\det A}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ — swap the leading diagonal, change the sign of the other two. (1)

    Since $\det A=1$, dividing changes nothing: $A^{-1}=\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$. Check: $A{A}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. (1)

    Final answer: $\det A=1$ and $A^{-1}=\left[\begin{array}{cc}3& -5\\ -1& 2\end{array}\right]$.

21. Question 21

    1 mark

    What is the transpose of $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$?

    A. $\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

    B. $\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right]$

    C. $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

    D. $\left[\begin{array}{cc}4& 3\\ 2& 1\end{array}\right]$

    (1 mark)

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