Year11 MATH 2-2-5 Warm-up Questions-Algebra and Matrices (The Logic of Arrays)

In Chapter 5 of General Mathematics, we step into the world of Matrices. These are not just boxes of numbers; they are powerful tools for organizing data, solving systems of equations, and analyzing complex networks. Mastering the “Logic of Arrays” allows you to process large amounts of information simultaneously—a skill essential for computer science, economics, and engineering.

Concepts and Skills Covered:

1. Matrix Notation: Identifying the order (rows $\times$ columns) and specific elements ($a_{ij}$).
2. Arithmetic Operations: Performing addition, subtraction, and scalar multiplication.
3. Matrix Multiplication: Understanding conformability and the row-by-column multiplication process.
4. Special Matrices: Utilizing Identity ($I$) and Zero ($O$) matrices.
5. $2 \times 2$ Determinants and Inverses: Calculating the determinant ($ad – bc$) and finding the inverse to solve equations.Network Analysis: Representing connections as adjacency matrices and using powers of matrices (like $A^2$) to find multi-step paths.

***************************

Q1. What is the order (dimensions) of the matrix $A = \begin{pmatrix} 4 & -1 & 0 \\ 2 & 5 & 8 \end{pmatrix}$?

A. 6

B. 2×3

C. 1×6

D. 3×2

Hint: Count the number of horizontal rows first, then the number of vertical columns.

Q2. Under what condition can two matrices, $A$ and $B$, be added or subtracted?

A. They must have the exact same order (dimensions).

B. The determinant of both matrices must be non-zero.

C. The number of columns in A must equal the number of rows in B.

D. They must both be square matrices.

Hint: Think about whether you could add a $2 \times 2$ matrix to a $3 \times 3$ matrix.

Q3. If $B = \begin{pmatrix} 3 & -2 \\ 1 & 4 \end{pmatrix}$, what is the result of the scalar multiplication 3B?

A. $\begin{pmatrix} 9 & -2 \\ 1 & 4 \end{pmatrix}$

B. $\begin{pmatrix} 27 & -18 \\ 9 & 36 \end{pmatrix}$

C. $\begin{pmatrix} 9 & -6 \\ 3 & 12 \end{pmatrix}$

D. $\begin{pmatrix} 6 & 1 \\ 4 & 7 \end{pmatrix}$

Hint: Distribute the number 3 to every number inside the brackets.

Q4. Matrix $M$ has order $3 \times 2$ and Matrix $N$ has order $2 \times 5$. What is the order of the resulting matrix $MN$?

A. The multiplication is not possible.

B. 5×3

C. 2×2

D. 3×5

Hint: The ‘inner’ numbers must be the same to multiply, while the ‘outer’ numbers determine the new shape.

Q5. Given $A = \begin{pmatrix} 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, calculate the product $AB$

A. $\begin{pmatrix} 7 \end{pmatrix}$

B. $\begin{pmatrix} 3 & 8 \end{pmatrix}$

C. $\begin{pmatrix} 11 \end{pmatrix}$

D. $\begin{pmatrix} 3 & 4 \\ 6 & 8 \end{pmatrix}$

Hint: Multiply the row elements by the column elements and add them together.

Q6. What is the result when any square matrix $A$ is multiplied by the identity matrix $I$ of the same order?

A. Matrix $A$

B. The Inverse matrix $A^{-1}$

C. The Identity matrix$I$

D. The Zero matrix O

Hint: Consider the property of the ‘identity’ in regular arithmetic (e.g., $5 \times 1$).

Q7. Calculate the determinant of the matrix $C = \begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}$

A. 7

B. 14

C. −14

D. 26

Hint: Subtract the product of the ‘off-diagonal’ from the product of the ‘main diagonal’.

Q8. A $2 \times 2$ matrix is said to be ‘singular’ (having no inverse) if:

A. All of its elements are positive.

B. It contains at least one zero element.

C. Its determinant is equal to zero.

D. It is not a square matrix.

Hint: Think about why you cannot divide by zero in the inverse formula.

Q9. In an adjacency matrix representing a network, what does the entry in row $i$, column $j$ typically represent?

A. The number of two-step paths between the vertices.

B. The number of direct paths (one-step) from vertex i to vertex j.

C. The importance or ‘weight’ of the vertex.

D. The total distance between vertex i and vertex j.

Hint: An adjacency matrix is a map of ‘who is connected to whom’.

Q10. If $A$ is an adjacency matrix for a communication network, what information does the matrix $A^2$ provide?

A. A list of vertices that are not connected to any others.

B. The square of every individual number in the original matrix.

C. The number of 2-step paths between any two vertices.

D. The network with all connection directions reversed.

Hint: Think about what happens to ‘steps’ in a path when you multiply the connection matrix by itself.

*********

一般数学の第5章では、行列の世界に触れます。行列は単なる数字の箱ではありません。データの整理、連立方程式の解法、複雑なネットワークの解析に役立つ強力なツールです。「配列の論理」を習得することで、大量の情報を同時に処理できるようになります。これは、コンピュータサイエンス、経済学、工学に不可欠なスキルです。

扱う概念とスキル：

行列の表記：順序（行 $\times$ 列）と特定の要素（$a_{ij}$）を識別する。

算術演算：加算、減算、スカラー乗算を実行する。

行列の乗算：適合性と行と列の乗算プロセスを理解する。

特殊行列：単位行列（$I$）と零行列（$O$）を活用する。

$2 \times 2$ 行列式と逆行列：方程式を解くために行列式 ($ad – bc$) を計算し、逆行列を求める。

ネットワーク分析：接続を隣接行列として表現し、行列のべき乗（$A^2$ など）を用いて多段階経路を求める。

**************************

Right answers.

Q1. B. $2 \times 3$

The order is written as rows $\times$ columns; this matrix has 2 horizontal rows and 3 vertical columns.

Q2. A. They must have the exact same order (dimensions).

Matrix addition is performed element-by-element, so the positions must correspond exactly in both matrices.

Q3. C. $\begin{pmatrix} 9 & -6 \\ 3 & 12 \end{pmatrix}$

Scalar multiplication involves multiplying every individual element inside the matrix by the constant 3.

Q4. D. $3 \times 5$

The inner dimensions (2) match for multiplication, and the outer dimensions (3 and 5) define the size of the result.

Q5. C. $\begin{pmatrix} 11 \end{pmatrix}$

Multiplying a $1 \times 2$ by a $2 \times 1$ gives a $1 \times 1$ result: $(1 \times 3) + (2 \times 4) = 3 + 8 = 11$.

Q6. A. Matrix $A$

The identity matrix acts like the number 1 in scalar multiplication; it leaves the original matrix unchanged ($AI = IA = A$).

Q7. B. 14

The determinant is calculated as $ad – bc$, which is $(5 \times 4) – (2 \times 3) = 20 – 6 = 14$.

Q8. C. Its determinant is equal to zero.

Since the formula for the inverse involves dividing by the determinant, a zero value makes the calculation undefined.

Q9. B. The number of direct paths (one-step) from vertex $i$ to vertex $j$.

Adjacency matrices record the immediate connections between nodes in a graph or network.

Q10. C. The number of 2-step paths between any two vertices.

Raising the adjacency matrix to the power of $n$ reveals the number of $n$-step paths available in the network.